On the Out?of?Plane Vibration of Rotating Circular Nanoplates

，，*，

1.School of Rail Transportation，Soochow University，Suzhou 215131，P.R.China；

2.School of Architectural Engineering，Tongling University，Tongling 244061，P.R.China

Abstract: A rotating axisymmetric circular nanoplate is modeled by the Mindlin plate theory. The Mindlin plate theory incorporates the nonlocal scale and strain gradient effects. The shear deformation of the circular nanoplate is considered and the nonlocal strain gradient theory is utilized to derive the governing differential equation of motion that describes the out-of-plane free vibration behaviors of the nanoplate. The differential quadrature method is used to solve the governing equation numerically，and the natural frequencies of the out-of-plane vibration of rotating nanoplates are obtained accordingly. Two kinds of boundary conditions are commonly used in practical engineering，namely the fixed and simply supported constraints，and are considered in numerical examples. The variations of natural frequencies with respect to the thickness to radius ratio，the angular velocity，the nonlocal characteristic scale and the material characteristic scale are analyzed in detail. In particular，the critical angular velocity that measures whether the rotating circular nanoplate is stable or not is obtained numerically. The presented study has reference significance for the dynamic design and control of rotating circular nanostructures in current nano-technologies and nano-devices.

Key words：circular nanoplate；nonlocal strain gradient；differential quadrature method；material characteristic scale；angular velocity

0 Introduction

With the rapid development of nanotechnolo?gy，the materials and structures at a nanoscale have attracted increasing attentions during the past two decades［1-3］. As a new research field，nanotechnolo?gy involves the property，optimization and applica?tion of materials with scales between 1 nm and 100 nm，including nanoelectronics，nanophotonics，nanobiology，nanomedicine and other branch sub?jects，among which the mechanical property of nanomaterials and nanostructures，i.e.，the nanome?chanics plays an important role. The mechanical characterizations with related nanoscale properties and parameters are indispensable in the preparation，testing，modification and optimization of nanomate?rials［4］. Stationary nanomaterials and nanostructures are currently characterized and tested easily but re?quires a static condition. Performance characteriza?tion under motion is one of the technical difficulties，which hinders the development of new nanomateri?als and functional structures based on kinetic de?sign［5］. Therefore，it has important scientific signifi?cance in studying the mechanical properties of nano?materials and nanostructures，especially the dynam?ic properties of those with motion. Regarding the re?search methodologies，theoretical approaches and corresponding models and analyses are more popu?lar in view of the complexity of manipulation and testing of nanoscale dynamic experiments［6］. The theoretical approaches are mainly divided into two categories. One is the discrete atomic model［7］. By regarding nanomaterials as an integration of a cer?tain number of atoms，the discrete model of nano?materials is established by considering the forces be?tween atoms. The other is the non-classical continu?um model［8］that regards nanomaterials as a general?ized continuum and accounts for the internal charac?teristic scale parameters of materials. However，when the number of atoms compose nanomaterials and nanostructures is relatively large，the discrete model is cumbersome in both modeling and calcula?tion. For instance，the calculation costs extra hard?ware and long time. With the requirements of ana?lyzing complex nanosystems，the non-classical con?tinuum model dominates the theoretical prediction of dynamic phenomena in nanomaterials and nano?structures.

For non-classical continuum models，since the beginning of the last century，the couple stress theo?ry and its modified versions，the micropole and mi?crostate theory，the strain gradient theory，the nonlocal theory，and the nonlocal strain gradient the?ory have been proposed successively［9］. These theo?retical methods have been applied to the analysis of mechanical properties of nanostructures during past years. For instance，Akg?z et al.［10］proposed a sizedependent higher-order shear deformation beam model based on the Navier method and the modified strain gradient theory in which both the microstruc?tural and shear deformation effects were taken into consideration. Akg?z et al.［11］studied the thermomechanical buckling behaviors of embedded func?tionally graded microbeams based on the sinusoidal shear deformation beam theory and modified couple stress theory. Numano?lu et al.［12］investigated the longitudinal dynamic properties of nanorods with various boundary conditions based on the nonlocal theory. Ebrahimi et al.［13］developed a nonlocal cou?ple stress theory to reveal the vibration behaviors and stabilities of functionally graded nanobeams us?ing the Chebyshev-Ritz method. Nevertheless，there are still some unsolved problems in the theo?retical application. For example，the micro-soften?ing and hardening predictions were contradicted［14］，and the undefined or inconsistent internal character?istic scale parameters［15］were used in previous stud?ies. In regard of these，Lim et al.［16］established a nonlocal strain gradient theory in 2015，which intro?duced and coupled both the nonlocal parameter and material characteristic scale parameter to measure the nonlocal effect and strain gradient effect of nano?materials and structures，respectively. Accordingly，the total nonlocal strain gradient stress is defined，and the constitutive relations reflect the nonlocal ef?fect of classical strains and strain gradients，as well as the gradient effect of nonlocal stresses and total stresses，which promotes the application adaptabili?ty of the nonlocal strain gradient theory at a na?noscale. Meanwhile，it has been proved that the mi?cro-softening and hardening phenomena are ob?served and in fact they are described by two special cases of the theory，namely，corresponding to two simplified forms of the theory［17］. Moreover，it also has the guiding significance for the determination of the internal characteristic scale parameters in the non-classical continuum theory［18］. Consequently，the nonlocal strain gradient theory is suitable for the study of nanomechanics. This is why it has became a popular research method in nanomechanics since the theory was put forward［19-24］. In this paper，the nonlocal strain gradient theory is used to examine the vibration behavior of circular nanoplates.

The vibration and stability of axially moving nanostructures have been fully studied during the past several years［25-30］. However，rotating nano?structures are relatively less studied. The rotating circular nanoplate is one of the important compo?nents in the nanoelectromechanical system［31］that is usually used to realize power and motion transmis?sions. Hence it is a common structure for the system operation. To solve the vibration equations to show dynamic behaviors and other related characteristics of the circular nanoplates，the differential quadra?ture method and the nonlocal finite element method are usually used［32-34］. Generally，the structural char?acteristics，forces and constraints of circular nano?plates are axisymmetric. Therefore，the present work concerns the free vibration characteristics of axisymmetric circular nanoplates with rotational mo?tion. The results may provide a theoretical basis for the dynamic design and optimization of key compo?nents in the nanoelectromechanical system.

1 Theory and Method

1.1 Nonlocal strain gradient theory

The nonlocal strain gradient theory combines the higher-order stress gradients with the nonlocal effect of strain gradients. The constitutive relations can be expressed as［16］

whereσandσ(1)represent the nonlocal stress tensor and the higher-order nonlocal stress tensor in vol?umeV，respectively；α0andα1the nonlocal kernel functions related to the strain and the first-order strain gradient；ande0ande1the traditional nonlocal and higher-order nonlocal material constants.trepre?sents the nonlocal strain gradient total stress tensor；Cthe elastic tensor；ε′the strain tensor at pointx′；athe nonlocal characteristic scale；lthe material characteristic scale related to higher-order strain gra?dients；and ?the gradient operator. As a result，the nonlocal stress and higher-order nonlocal stress at pointxdepends on not only the strain and strain gra?dient at pointx，but also the strain and strain gradi?ent at pointx′in the nonlocal strain gradient theory.That is，the idea of long-range interactions between molecules/atoms is introduced into both the nonlo?cal and strain gradient constitutive relations.

The above integral constitutive equations are difficult to solve. Fortunately，they can be trans?formed into differential constitutive equations.Based on certain assumptions，the core constitutive relation of the nonlocal strain gradient theory can be simplified by reorganizing the integral constitutive equations and introducing the Laplace operator as［16］

where ?2is the Laplace operator，and two material constants are assumed to be identical，i.e.e0=e1=e.

1.2 Differential quadrature method

There are different methods to solve differen?tial equations，among which the differential quadra?ture method has been widely used due to its fast cal?culation and high accuracy. In this numerical meth?od，the function values of all nodes in the whole do?main are weighted and summed to represent the function value and its derivatives at the selected node. Resultingly，the differential equations can be discretized into a set of algebraic equations with the node values as the unknown variables. For a one-di?mensional functionf（x），let it be continuously dif?ferentiable in the interval［a，b］，and one obtains

whereLis a linear differential operator；Wm（x）the interpolation basis function；andxmthemth node in the interval［a，b］.

then

wherej=1，2，…，N，andC(1)jmis the first-order weighting coefficient of functionf（x）. Accordingly，is the weighting coefficient matrix of its first derivative.

Denotef(k)j=f(k)(xj)，then the higher-order derivative at the function node can be represented by the interpolation of function values as

To determine the weight coefficients， letf(x)=xk-1，the following formula can be obtained from Eq.（4）as

Eq.（8）can be written in a matrix form asG=CV，where

Lagrange interpolation can be used as the ex?pression of a function node

wherelm（x）is the Lagrange interpolation polynomi?al as

The first derivative of Eq.（11）is

Consequently

Comparing Eq.（4）and Eq.（14），we can get

Therefore，the explicit expressions of the firstorder and higher-order weight coefficients can be de?termined. Furthermore，a reasonable node distribu?tion should be selected while using the differential quadrature method. The previous studies［32］show that the roots of Chebyshev polynomial can make the calculation faster with more accurate. There?fore，it is adopted in the present study as

2 Problem Model and Governing Equation

Considering an axisymmetric circular nanoplate rotating at an angular velocityΩ，with a radiusRand thicknessh，we establish a polar coordinate sys?tem，as shown in Fig.1，wherez-coordinate is along the axis（i.e. thickness direction）of the circu?lar nanoplate.

Fig.1 Diagrammatic sketch of an axisymmetric rotating cir?cular nanoplate

Based on the Mindlin plate theory，the radial displacementurand lateral displacementuzof the cir?cular nanoplate can be expressed as

wherew（r，t）represents the lateral displacement of any point on the midplane；φ（r，t）the rotational an?gle of the midplane normal；andtthe time.

From Eq.（17），the geometric equation can be obtained as Considering Eq.（18）together with the physi?cal equation，one can obtain the classical stress com?ponents as

whereεr，εθandγrzare the radial strain，the hoop strain and the shear strain，respectively；σr，σθandτrzthe radial stress，the hoop stress and the trans?verse shear stress，respectively；andEandνthe elastic modulus and the Poisson’s ratio，respective?ly. Subsequently，the internal forces including the axial force，the bending moment and the transverse shear force can be obtained by integrating Eqs.（19—21）. Therefore，the constitutive equations based on the nonlocal strain gradient theory can be derived

whereNrandNθare the components of the axial force；andMrandMθthe components of the bend?ing moment.Qris the transverse shear force，andκ=12/π2the shear correction factor［35］.

The first-order variation of the strain energyUof the rotating axisymmetric circular nanoplate can be calculated as

whereVrepresents the volume occupied by the cir?cular nanoplate. The first-order variation of the ki?netic energyTcan be determined as

whereρis the bulk density of the circular nanoplate.The variation of the potential energyHcaused by the rotation is

whereNRis the radial tension caused by the rotation.For the fixed and simply supported boundary con?straints，NRcan be derived as［33］

Based on Hamilton’s principle，the equation of motion that governs the free out-of-plane vibration of rotating axisymmetric circular nanoplates can be derived as

Substituting Eqs.（22—26）into Eqs.（36，37），the vibration governing equation of the rotating axi?symmetric circular nanoplate in the framework of the nonlocal strain gradient theory is

The classical out-of-plane vibration model of a rotating circular plate is recovered in case ofea=l=0.The dimensionless quantities are introduced

After that，dimensionless forms of Eqs.（38，39）can be obtained as

The solutions of Eqs.（41，42）can be set as

where?() and() are the vibration mode func?tions andωis the non-dimensional natural frequency of the free out-of-plane vibration.

Substituting Eq.（43）into Eqs.（41，42），one gets

The boundary conditions are

Using the differential quadrature method，we can discretize Eqs.（44，45）as

wherej=2，3，…，N-1.

Combining Eqs.（47，48） and the boundary conditions，the characteristic equation in matrix form can be written as

whereK，Mare the stiffness matrix and the mass matrix，respectively. Subscriptdrepresents the gov?erning equations；subscriptbthe boundary condi?tions；andqthe nodal displacement including the lateral displacementq1and the rotational angleq2.Therefore，K，Mare the coefficient matrices with respect to the lateral displacement and the rotational angle. The elements of the matrix consist of differ?ent equations that are associated with parametersζ，band so on. The expression of each element in the matrix is rather lengthy and is not specifically listed here.

3 Results and Discussion

In order to verify the effectiveness of the pro?posed calculation method，a simplified case of the present model，that is，the free vibration of a nonrotating circular macro-plate，is discussed herein.Letτ=ζ=λ=0. We can determine the first three natural frequencies of the circular nanoplate with dif?ferent ratios of thickness to radius under the fixed and simply supported boundary conditions according to Eqs.（47，48），which are compared with the re?sults available in Ref.［34］，as shown in Tables 1，2.

Mode b=0.05 The presented 10.143 38.852 84.992 Ref.[34]10.145 38.855 84.995 b=0.1 The presented 9.940 6 36.473 75.661 Ref.[34]9.940 8 36.479 75.664 b=0.15 The presented 9.628 3 33.392 65.547 Ref.[34]9.628 6 33.393 65.551 b=0.2 The presented 9.240 0 30.208 56.677 1 2 3 Ref.[34]9.240 0 30.211 56.682

Table 2 Comparison of the first three natural frequencies of the circular plate under the simply supported boundary condition

From Tables 1，2，the presented results are very close to those in Ref.［34］. Accordingly，the solution method and the numerical results are vali?dated. Besides，with an increase of the thickness to radius ratio（e.g. fromb=0.05 tob=0.2），there is a decrease in the natural frequency. Note that reduc?ing of the natural frequency means the nanostructur?al stiffness weakening. However，increasing the thickness to radius ratio corresponds to increasing the thickness or decreasing the radius. Increasing the thickness means increasing the nanostructural stiffness，while decreasing the radius means decreas?ing the nanostructural stiffness. As from the two as?pects，one can infer that the stiffness weakening ef?fect caused by decreasing the radius is greater than the stiffness enhancement effect caused by increas?ing the thickness. This is because only in this way the stiffness of the circular nanoplate will eventually reduce. Moreover，the natural frequencies under fixed support conditions are higher than those under simply supported ones.

Considering a circular nanoplate made of ce?ramics（Si3N4）with the thicknessh=2.5 nm，radi?usR=50 nm，densityρc=2 370 kg/m3，elastic modulusEc=348.43 GPa，and Poisson’s ratioν=0.3. NodeN=18 is selected in the numerical calcu?lations，and we examine the vibration characteristics of the rotating axisymmetric circular nanoplate with the fixed and simply supported outer boundary con?straints.

The effect of the thickness to radius ratiobon the first three dimensionless natural frequencies of the circular nanoplate is shown in Fig.2，whereea=0.5 nm，l=1 nm，τ=0.01 andζ=0.02. The natu?ral frequency decreases with the increase of the thickness to radius ratio，and the higher-order vibra?tion is affected more obviously by the change of the thickness to radius ratio.

Fig.2 Influence of the thickness to radius ratio on natural frequencies (τ=0.01, ζ=0.02, λ=0)

The natural frequencies versus the angular ve?locityλ，the nonlocal characteristic scale parameterea，and the material characteristic scale parameterlare shown in Figs.3—6. The nonlocal and material characteristic scale parameters represent the nonlo?cal and strain gradient effects，respectively，and are selected as zero in Fig.3，and the out-of-plane vibra?tion of the rotating axisymmetric circular plate or the classical model can be recovered. From Fig.3（a），it can be seen that natural frequencies with the periph?eral fixed boundaries decrease with the increasing angular velocity. When the angular velocity increas?es toλ1=15.689，the first natural frequency be?comes zero which indicates that the vibration of the rotating circular plate appears divergent instability.Soλ1=15.689 is called the first-order critical angu?lar velocity. Similarly，λ2=30.307 is the second-or?der critical angular velocity. From Fig.3（b），one can see that the first natural frequency of the periph?eral simply supported case does not decrease monot?onously with the increase of the angular velocity，which is different from the peripheral fixed con?straint，and the boundary effect in the out-of-plane vibration of the rotating axisymmetric circular plate is thus reflected. The instability characteristics are the same as those of the peripheral fixed rotating cir?cular nanoplate.

Fig.3 Influence of the dimensionless angular velocity on natural frequencies (τ=ζ=0)

Fig.4 Influence of the dimensionless angular velocity on natural frequencies (τ=0.04, ζ=0)

Fig.5 Influence of the dimensionless angular velocity on natural frequencies (τ=0, ζ=0.02)

Fig.6 Influence of the dimensionless angular velocity on natural frequencies (τ=0.04, ζ=0.02)

Comparing the results from Figs.3—6，we find that the natural frequency decreases with the increas?ing nonlocal characteristic scale parameter. The de?crease of natural frequencies means a stiffness weak?ening of structures. Hence it is demonstrated that the nonlocal characteristic scale has a softening ef?fect on the nanostructure. In Fig.4（a），the first-or?der critical angular velocity decreases from 15.689 to 15.324，and the second-order decreases from 30.307 to 27.626 with the increase of the nonlocal characteristic scale parameter. This means the nonlocal characteristic scale makes the critical angu?lar velocity decrease. In fact，such an observation is also one of the manifestations of the nonlocal soften?ing effect. Fig.5（a）shows that the natural frequen?cy increases with the increase of the material charac?teristic scale parameter，so the strain gradient has a hardening effect on the nanostructure. With the in?crease of the material characteristic scale parameter，the first-order and the second-order critical angular velocities increase from 15.689 to 15.783 and from 30.307 to 31.046，respectively. This indicates that the existence of the material characteristic scale in?creases the critical angular velocity and the strain gradient hardening effect is revealed again.

To reveal the relationship between the nonlocal scale and strain gradient effects in the out-of-plane vibration of the rotating axisymmetric circular nano?plate，we display the effect of the ratioea/lon the first three natural frequencies，as shown in Fig.7.As observed，the natural frequency decreases with the increase ofea/l. As mentioned before，the theo?retical model will degenerate to the classical counter?part wheneaandlare both equal to zero. In fact，this condition can be relaxed. As long aseais equal tol，even if both are not zero，the natural frequen?cies keep unchanged with respect to the ratioea/l.Consequently，the results are always equal to those of the classical counterpart，and both the softening and hardening phenomena disappear，which shows that the softening/hardening effects derived from the two internal characteristic scale parameters on the nanostructures can cancel each other. This im?plies that the nonlocal scale effect and strain gradi?ent effect have the opposite mechanisms in nano?structures，but the degree of effects is equivalent.Whenea/lis less than 1，that is，eais less thanl，the natural frequencies based on the nonlocal strain gradient theory is greater than those based on the classical vibration theory，and the larger the materi?al characteristic scale parameter，the greater the de?gree of deviation from the classical results. In this in?stance，the characteristic scale parameters strength?en the stiffness of the rotating circular nanoplate.However，whenea/lis greater than 1，that is，eais greater thanl，the natural frequency based on the nonlocal strain gradient theory is less than their clas?sical counterparts，and the larger the material char?acteristic scale parameter is，the more obvious this phenomenon is. So the internal characteristic scale parameters weaken the nanoplate stiffness. There?fore，there is a coupling relationship between the nonlocal and material characteristic scale parame?ters. The magnitude relationship between the two internal characteristic scale parameters will deter?mine the specific manifestation of the internal scale effects in the nonlocal strain gradient theory，that is，the nonlocal softening or strain gradient harden?ing phenomenon in the rotating circular nanoplate.When the nonlocal characteristic scale parameter is larger than the material characteristic scale parame?ter，the nonlocal softening effect plays a dominated role in the nonlocal strain gradient theory. Other?wise，the strain gradient hardening effect dominates the internal characteristic scale effects

Fig.7 Relationship between ea/l and the dimensionless natural frequency (λ=0)

4 Conclusions

The out-of-plane vibration analyses of the rotat?ing axisymmetric circular nanoplate is carried out based on the nonlocal strain gradient theory and the Mindlin plate model. The governing differential equation of motion that includes the nonlocal and material characteristic scale effects is derived，and the natural frequency is calculated by the differential quadrature method. The natural frequency decreases with the increase of the thickness to radius ratio，and the higher-order frequencies are influenced more significantly. The natural frequency decreases with the increase of the angular velocity. When the angu?lar velocity increases to a critical value，the natural frequency becomes zero. The critical angular veloci?ty is greatly affected by the internal characteristic scale parameters.The existence of the nonlocal char?acteristic scale parameter reduces the critical angular velocity，but the existence of the material character?istic scale parameter causes it to increase. The stiff?ness of the circular nanoplate decreases with the in?crease of the nonlocal characteristic scale parame?ter，but increases with the increase of the material characteristic scale parameter. The two internal scale parameters are coupled in the nonlocal strain gradient theory. The parameter with a larger value is dominant in the internal characteristic scale ef?fects.

AcknowledgementsThis work was supported by the Natu?ral Science Foundation of China（No.11972240），the China Postdoctoral Science Foundation（No.2020M671574），and the University Natural Science Research Project of Anhui Province（No.KJ2018A0481）.

AuthorsMs. WANG Xinyue received her bachelor degree in Vehicle Engineering from Soochow University. She is cur?rently a postgraduate of the School of Rail Transportation，Soochow University，Suzhou，China. Her research is fo?cused on metamaterials，smart structures and their applica?tions in vehicle dynamics and control.

Prof. LI Cheng is a professor of the School of Rail Transpor?tation, Soochow University, Suzhou, China. He received his Ph.D degree in Engineering Mechanics from Department of Architecture and Civil Engineering, City University of Hong Kong. His research interests include structural mechan?ics, nonlinear vibration of engineering structures, and vehi?cle dynamics and control.

Author contributionsMs. WANG Xinyue designed the study，analyzed the results，investigated literature and wrote the manuscript. Mr. LUO Qiuyang conducted calculation，drew the image，and collated the results. Prof. LI Cheng was responsible for supervision and provided methods and resources. Prof. XIE Zhongyou 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 interests.