Thermo-mechanical stability of single-layered graphene sheets embedded in an elastic medium under action of a moving nanoparticle

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Thermo-mechanical Stability of Single-layered Graphene Sheets Embedded in an Elastic Medium under Action of a Moving Nanoparticle Mostafa Pirmoradian , Ehsan Torkan , Nasir Abdali , Mohamad Hashemian , Davood Toghraie PII: DOI: Reference:

S0167-6636(19)30581-2 https://doi.org/10.1016/j.mechmat.2019.103248 MECMAT 103248

To appear in:

Mechanics of Materials

Received date: Revised date: Accepted date:

6 July 2019 14 November 2019 15 November 2019

Please cite this article as: Mostafa Pirmoradian , Ehsan Torkan , Nasir Abdali , Mohamad Hashemian , Davood Toghraie , Thermo-mechanical Stability of Single-layered Graphene Sheets Embedded in an Elastic Medium under Action of a Moving Nanoparticle, Mechanics of Materials (2019), doi: https://doi.org/10.1016/j.mechmat.2019.103248

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Highlights 

A nonlocal thermo-mechanical model for linear instability analysis of single-layered Graphene sheets



Analysis in thermal environment using an energy-based method



All inertial effects of the moving nanoparticle are taken into account



1

Thermo-mechanical Stability of Single-layered Graphene Sheets Embedded in an Elastic Medium under Action of a Moving Nanoparticle Mostafa Pirmoradian1, Ehsan Torkan2, Nasir Abdali1, Mohamad Hashemian1, Davood Toghraie1 1

Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr/Isfahan, 84175–119, Iran 2 Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr/Isfahan, 84175–119, Iran

*

Corresponding author: Davood Toghraie, Department of Mechanical Engineering, Islamic Azad University, Khomeinishahr Branch, Khomeinishahr 84175-119, Iran. Email: [email protected] Abstract Using an energy-based method, this paper sought to analyze dynamic stability and parametric resonance of single-layered graphene sheets (SLGSs) embedded in thermal environment and elastic medium while carrying a nanoparticle moving along an elliptical path. In order to present a realistic model, all inertial effects of the moving nanoparticle are taken into account in the dynamic formulation of the system. Equations governing the transverse vibrations of the embedded SLGS are obtained using the Hamilton’s principle. Small-scale effects based on the Eringen's nonlocal elasticity theory are considered in deriving the motion equations. The equations governing the reduced model are calculated based on the Galerkin method. To calculate the instability boundaries, the energy-rate method is applied on the ordinary differential equations (ODEs) governing the system oscillations. The effects of nonlocal parameter, the nanoparticle motion path radii, SLGS length-to-width ratio, temperature change of the thermal environment, stiffness of the elastic medium and boundary conditions of SLGS on the parametric instability regions are examined. The results show that these parameters influence the system stability, so that a decrease in the nonlocal parameter, the SLGS length-to-width ratio and the nanoparticle motion path radii and also an increase in the stiffness coefficients of the elastic medium improve the system stability. The model presented in this paper is validated by comparing the observations with those published in previous studies.

Keywords: Graphene sheets, Moving nanoparticle, Dynamic stability, Thermal effects, Energy-rate method.

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1. Introduction Wonderful thermal, mechanical and electrical properties of nanostructures such as nanotubes [1], nanowires [2], graphene sheets [3] seem to hold a great promise to use them for a wide range of applications known as nano/micro-electro-mechanical systems. Graphene as the thinnest two-dimensional nanostructure, due to its unique properties, is considered by many researchers. Recent mechanical experiments have shown that graphene, having an elastic modulus of 1 TPa [3], is the strongest ever known material. Graphene is the best option for designing and developing nano-electro-mechanical systems (NEMSs) such as high speed transistors [4], solar cells [5] and batteries [6]. In the analysis of nanoscale structures, conducting experimental researches is difficult and molecular simulations are costly and time-consuming. As a result, development of mathematical modeling is used to evaluate the mechanical responses of nanostructures. Perhaps the continuum mechanics modeling is the most efficient method to model nanostructures. The classical continuum model is unable to compute quantum effects caused by discrete nature of nanoscale materials. At the nanoscale, the size of particles is comparable with the interatomic or intermolecular distances and thus the system cannot be modeled as a continuum. This matter is called as the size effect feature which leads to interesting and unconventional behavior of nanostructures. Results of simulations show that at nanometeric scales, the size effect usually has a significant impact on the analysis of nanostructures. Reducing the longitudinal scales will enhance the importance of the effect of coherent intermolecular and interatomic forces on the dynamic and static properties of nanostructures. Hence, to overcome the classical mechanics limitations, different modified continuum models have been proposed including Eringen’s nonlocal elasticity theory [7], strain gradient theory [8], coupled stress theory [9], and modified coupled stress theory [10]. Eringen’s nonlocal

3

elasticity theory is widely used in examining the small-scale effect. Hence, many articles have used this theory to analyze nanostructure problems. Aghababaei and Reddy [11] used the nonlocal elasticity theory to analyze nanoplates. They extracted the nanoplate motion equations based on the nonlocal third-order shear deformation theory and examined effect of nonlocal parameter on the natural frequencies of simply supported nanoplates. Pradhan and phadikar [12] used the nonlocal elasticity theory to examine linear free vibration of single-layered and double-layered classic isotropic nanoplates with simply supports. Taking into account the classical plate theory and the nonlocal elasticity theory, Shen et al. [13] examined the nonlinear free vibration of a singlelayer orthotropic nanoplate embedded in a temperature change environment. In other study, Ansari et al. [14] investigated vibrations of multi-layered graphene sheets (MLGSs) by considering the Van der Waals effects for different boundary conditions. In a similar study by Ansari et al. [15], nonlocal finite element model was presented to calculate natural frequencies of multi-layered graphene nanoplates. In their work, Aksencer and Aydogdu [16] studied the forced transverse vibrations of simply supported single-layered isotropic nanoplates based on the classical plate theory along with the nonlocal elasticity theory. In another study, based on the nonlocal elasticity theory, Wang et al. [17] examined effect of temperature on the free vibration of double-layered isotropic nanoplates. A study by Shi et al. [18] dealt with wave propagation characteristics of double-layered graphene sheets (DLGSs) based on the nonlocal Mindlin plate theory. Furthermore, Nazemnejad et al. [19] examined effect of nonlocal elasticity on the vibrations of MLGSs. To validate the employed model, they used the molecular dynamics (MD) simulation. The problem of nanoplates carrying moving nanoparticles is one of the most widely used problems in the nanotechnology. For example, NEMSs such as atomic dust detectors, mass

4

sensors and nanoswitches can be modeled as nanoplates carrying moving nanoparticles. Limited studies are available on dynamic analysis of nanoplates carrying moving nanoparticle [20–24]. Ignoring the inertial effects of the nanoparticle and taking into account the Coulomb friction, Kiani [20] studied nonlocal vibrations of simply supported elastic nanoplates under passage of a moving nanoparticle. Based on nonlocal elasticity theory, Kiani [21,22] calculated the dynamic response of a nanoplate carrying a moving nanoparticle. Considering the rectilinear and elliptical paths for the motion of the nanoparticle, the system equations were derived using Hamilton’s principle and based on the Kirchhoff, Mindlin and higher-order shear deformation theories. Also, Ghorbanpour Arani et al. [23] examined the dynamic response of polyvinylidene fluoride nanoplates carrying a nanoparticle moving on an elliptical path. Furthermore, Bakhshi Khaniki and Hosseini-Hashemi [24] investigated dynamic response of a double-layered viscoelastic orthotropic nanoplate under a moving nanoparticle. The couplings between layers were modeled using Kelvin-Voigt viscoelastic theory by the assumption that each layer was bearing a biaxial load. As can be seen, although the problem of interaction of nanoplates and moving nanoparticles is a practical question, enough consideration has not been attended to its stability characteristic. Motivated by this consideration, in order to improve optimum design of nanostructures, the present study analyzes dynamic stability and parametric resonance of SLGSs carrying a moving nanoparticle, which has not been addressed in previous studies. In order to take into account the more reasonable problem, the inertial effects of the nanoparticle also included in the companion paper. In addition, the elliptical path as a general motion trajectory was also considered for the moving particle. In mathematical modeling, the governing partial equations are derived using the Hamilton’s principle and small-scale effects are applied to equations based on the nonlocal elasticity theory. Then, the Galerkin method is used to reduce the partial differential equations (PDEs) to ordinary differential equations (ODEs). An

5

energy-based method (energy-rate method [25]), which has not been used in the stability analyses of nanostructures subjected to parametric excitations, is utilized to analyze the system stability. To validate the model presented in this article, results are compared with those of previous articles.

2. Nonlocal model of the SLGS carrying a moving nanoparticle One of the well-known theories of continuum mechanics, which includes small-scale effects with good accuracy and consistency with MD, is Eringen’s nonlocal elasticity theory [7]. The nonlocal elasticity theory is based on the assumption that stress at a reference point is not only a function of the strain field of that point, but also it depends on the strain field at other points of the body. Accordingly, the integral form of the constitutive relation for homogenous elastic bodies can be expressed as follows:

   V K  x  x ,  t  x dV  x  ,

(1)

t ( x)  C ( x) :  ( x),

(2)

where  and t  x  are the nonlocal and local stress tensors, respectively. The kernel function K  x  x ,  denotes a nonlocal modulus and x  x represents the Euclidean distance. The material constant  is defined as e 0a l where l and a denote external and internal characteristic length of the material, respectively. In addition, the term e0 denotes a material constant. Also, in Eq. (2) C ( x) and   x  are the fourth-order elasticity and strain tensors, respectively. It is often difficult to solve the elasticity problems by using the integral constitutive Eq. (1). So, a simplified constitutive relation in differential form, given by Eringen, is used as follows [7]:

6

1      t, 2

where 2  

(3)

2

x

2



2

y

2



2

parameter for which selecting

is the Laplacian operator, and    e0 a  is a nonlocal 2

z

2

appropriate value is vital for the nonlocal models. The

nonlocal constitutive relation of Eq. (3) can be easily solved in mathematics and is widely employed in nanomechanical problems such as bending, buckling, wave propagation and vibration of nanostructures. The system considered in this research, as shown in Fig. 1, includes a rectangular nanoplate of the length a, width b, thickness h, and embedded in an elastic medium. Mechanical properties of the nanoplate include the Young’s modulus E, Poisson’s ratio  and density ρ. The elastic medium is considered as a Pasternak foundation including a shear layer with stiffness G p and springs with stiffness k w . In addition, the system is embedded in a thermal environment with temperature changes T . Also, it is assumed that the nanoparticle m moves along an arbitrary path over the nanoplate’s surface.

Fig. 1 A nanoparticle moving on an arbitrary path over the nanostructure.

7

The relation of bending moment and stress in classical elasticity theory is reliable in the nonlocal elasticity, because both the classical and nonlocal elasticity approaches belong to continuum theories. These relations are defined as follows:

M

xx

, M xy , M yy  

h /2

 

xx

,  xy ,  yy  z dz,

(4)

 h /2

where M xx , M xy and M yy are the components of bending moment.

In this research, modeling of the nanoplate is based on the Kirchhoff's plate theory. According to this theory, the displacement field is defined based on the Kirchhoff’s assumptions as follows [26]: 1. Straight lines perpendicular to the middle surface of the plate (i.e. transverse normals) do not experience elongation. 2. The transverse normals before deformation remain straight after deformation. 3. The transverse normals rotate such that they remain perpendicular to the middle surface after deformation. Based on these assumptions, time-dependent deformations of the nanoplate, neglecting displacements of the middle surface along the directions x and y, are as follows:

w  x, y, t  , x w  x, y, t  u y  x, y , z , t    z , y u x  x, y , z , t    z

(5)

u z  x, y , z , t   w  x, y , t  ,

8

where u x , u y and u z are displacements of an arbitrary point of the nanoplate along x, y and z axes, respectively; and w is the transverse displacement of the nanoplate middle surface. For small deformations, the strains are introduced as follows:

u x , x u  yy  y , y u  zz  z , z u 1  u  xy   x  y 2  y x

 xx 

1  u

 xz   x 2  z 1  u y 2  z

 yz  

(6)

 ,  u   z , x  u   z , y 

The first and the second Kirchhoff’s assumptions imply that the transverse strain along the thickness of the nanoplate is equal to zero, that is:

 zz 

u z  0. z

(7)

Also, Kirchhoff’s third assumption shows that the transverse shear strains are equal to zero, which yields: u x u z   0, z x u y u z    0. z y

 xz   yz

(8)

Therefore, the strains for small deformations are as follows:

9

2w x 2 2w  yy   z 2 y

 xx   z

 xy   z

(9)

2w xy

By expanding Eq. (3) and based on the classical plane stress relation to introduce the classical stress tensor t , the stress-strain relations based on the nonlocal theory are derived as the following:

 E  2  xx   xx  1  2     2  xy   e 0a    xy    0       yy   yy   E  1  2

0 E 2 1   0

E   1  2    xx   0   xy  ,   yy   

(10)

E   1  2 

By substituting Eq. (9) into Eq. (10), and then employing Eq. (4) to obtain the correlations between the nonlocal bending moment and corresponding classical counterpart, we have   2w  2w M xx   2 M xx  D  2  y 2  x

 , 

  2w  2w  M yy   2 M yy  D  2  , x 2   y

(11)

 2w  M xy   2 M xy   D 1    ,  xy 





where D  Eh3 12 1  2 is the bending stiffness of the nanoplate. The strain energy function for the elastic nanoplate is as follows [26]:

10

 h /2           dz  xx xx xy xy yy yy   dx dy ,   h / 2 

 

1 U  2

A

(12)

Substituting Eqs. (10) into Eq. (12) and performing some algebraic operations, the strain energy of the nanoplate is obtained as

D U 2

 A

2   2   2 w   2 w  2 w   2   w   2 1      dx dy,  xy  x 2 y 2       

(13)

The kinetic energy function of the nanoplate can be stated as:

1 K 2

 A

 h /2 2 2     u  v  w2  dz  dx dy,     h /2 



(14)

Neglecting the displacement terms u and v and integrating over the nanoplate thickness, the kinetic energy will be obtained as:

K

1 2

  hw dx dy. 2

(15)

A

The virtual work caused by the reaction force resulting from motion of the moving nanoparticle can be expressed as follows: Wp  

 F  x, y, t  w  x, y, t  dx dy,

(16)

A

where F  x, y, t  is the external excitation function. Taking into account all acceleration terms of the moving nanoparticle, the aforementioned function can be written as follows [27–29]: 2 2   2 w  2 w  dx   2 w  dy   2 w  dx  dy   2 w  dx  F  x, y , t    m   g  2  2    2    2  2       t x  dt  y  dt  xy  dt  dt  xt  dt  

11

2

 2w  dy  y t  dt

 w   x

 d 2x  2  dt

 w   y

 d 2y  2  dt

     x  x m t    y  y m t   , 

(17)

where  . is the Dirac delta function which is used to determine the nanoparticle location on the nanostructure. As previously stated, the elastic medium is simulated using the Pasternak foundation. The work done by this foundation is written as follows [30]:

Wf 

1 2

  k w  G  w wdx dy, 2

w

(18)

p

A

Also, the work done due to temperature variations T is described as follows [31]:

1 WT  2

 A

2

 w w  NT    dx dy,  x y 

where N T 

h 2 h  2



(19)

E T T dz denotes the external force caused by temperature changing and 1 

T represents the thermal expansion coefficient. The Lagrangian of the system which is defined as the difference between kinetic energy of the system and its strain energies plus the work done on it is defined as follows: L  K  U  Wp  W f  WT 

(20)

The governing equations will be derived using the Hamilton’s principle by taking the time integral of the first Lagrangian variations as the folloing:

12

t2

  Ldt  0.

(21)

t 1

Substituting Eq. (20) into Eq. (21) beside considering Eqs. (13), (15), (16), (18) and (19) results in:

1    2 w 2  2 w  2 w   D  2 2 2     Fw   hw    w   2 1        xy  x 2 y 2   2 2   t1 A   2  w w   1 1   k w w  G p  2 w  w  NT     dx dy dt. 2 2  x y  

t2

 

(22)

Taking the first variations of the recent equation and then performing some algebraic operations, it concludes that: t2

 t1

A

 2 M xy  2 M yy   2 w  2 M xx 2    kw w  G p 2 w    h 2  2 2 t x xy y    2w 2w   NT  2  2   F   w dx dy dt  0, y   x 

(23)

In the recent equation, the bending moments are as follows:  2w 2w  M xx   D  2  2  , y   x  2w 2w  M yy   D  2  2  , x   y

(24)

2w M xy   D 1   . xy Integrating Eq. (23) by parts and setting coefficients of  w to zero, the following governing equation is obtained: 13

 2 M xy  2 M yy  2w 2w   2 M xx 2w 2  2    h  k w  G  w  N w  T  x2  y 2   F  0, p x 2 xy y 2 t 2  

(25)

Using Eqs. (11) and (25), the nonlocal PDE of the nanoplate surrounded by thermal environment and elastic medium while carrying a moving nanoparticle is obtained as follows:  4w  2w 2w  4w 4w  2w D  4  2 2 2  4   1    2   h 2  1    2  NT  2  2  x y y  t y   x  x  1   

2

 k

w  G p w   1     F . 2

w

(26)

2

Considering simply support and clamp boundary conditions for the nanoplate leads to following relations for edges of the nanoplate: 

Simply supported nanoplate

for x  0 and x  a : w  0,

(27)

M xxnl  0,

(28)

and for y  0 and y  b : w  0,

(29)

M yynl  0,

(30)



Clamped edges

For x  0 and x  a and also for y  0 and y  b :

w  0,

w  0. x

(31)

14

As can be seen, for clamped edges, all boundary conditions are of displacement type and therefore the displacement and the slope of the nanoplate can be computed like the classical theory. On the other hand, Eqs. (28) and (30) show that there exists a combination of displacement and stress boundary conditions for the simply supported edges. When   0 , the boundary conditions are reduced to those of classical theory and consequently the boundary conditions would just depend on the displacements. For this case, Eqs. (28) and (30) will change to  2w  0, x 2

(32)

 2w  0, y 2

(33)

For the case   0 , Eqs. (28) and (30) will result in  2w   2 M xx  , x 2 D x 2

(34)

2  2w   M yy  , y 2 D y 2

(35)

which arose a new problem; because to solve the classical form it is necessary to have the right side of Eqs. (34) and (35) equal to zero. In order to remove the ambiguity, one has to look for other auxiliary equations to determine the bending moment relationship with the nanoplate displacement components. Taking moments around x and y axes, we have: Qx 



h 2



 dz  h xz 2

h 2

M xy M x  3w I 2  , x t x y

(36)

M y

(37)

M xy  3w Q y  h  yz dz  I 2  , y t y x  2



where Q x

and Q y are the resultant shear stresses. Differentiating Eq. (36) with respect to

x results in:

15

 2 M xy  2 M xx Q x  4w  I 2 2  x 2 x t x x y

(38)

If under certain conditions the right side of Eq. (38) becomes zero, then it will be possible to convert the stress boundary conditions to the displacement boundary conditions and thus develop a classical solution. Now, we have to see in what conditions this is going to happen. To this end, the right side terms of Eq. (38) are analyzed. Three simplifications are used as follows: I)

According to Green-Lagrange strains definition, for edges of the nanoplate it can be written:

 xz  2 xz  

u w v w  , x x x y

(39)

 yz  2 yz  

u w v w  , y x y y

(40)

If nanoplate displacements in x and y directions are considered to be so small, the abovementioned and therefore the first right hand term of Eq. (38) will be neglected. II) To eliminate the second term of Eq. (38), it is sufficient to ignore the rotational inertia. III) To investigate the last term, it should be noted that in the classical solution and for the simply supported boundary condition, the boundary conditions in the corners are not taken into account for the torsional moment M xy . In fact, one can rewrite the last equation of Eq. (11) for simply supported edges of a nanoplate as the following: M xy  2 M xy  0.

(41)

Here, performing a simplification and modifying the boundary conditions so as to solve the problem, M xy is considered to be a constant, i.e. M xy  cte , and hence the last term of Eq.

16

(38) will vanish. Similar argument and calculations can be performed for nonlocal bending moment M yynl .

3. Discretizing the motion equation It is almost difficult to find a solution for the partial differential equation appearing in Eq. (26). Hence, the Galerkin approximate method is used to discretize the equation [26]. Accordingly, the variable w which is a function of the parameters x, y and t, is stated as follows: w  x, y , t  



  x,y  q t , i

(42)

i

i 1

where qi  t  is the time-dependent generalized coordinate for ith shape function i  x,y  of the plate which should be calculated. The shape functions have to be chosen such that fulfill essential boundary conditions of Eqs. (27-31). Here, the well-known shape functions for simply supported and clamped nanoplates without presence of a nanoparticle are selected as follows [32]:

i  x, y   i  x, y  

2  m x   n y  sin   sin  , ab  a   b 

(43)

2   2m x    2n y    cos  a   1 cos  b   1 , 3 ab       

(44)

where m and n are vibrational modes along the length and width of the nanoplate, respectively. Substituting Eq. (42) into Eq. (26), then multiplying both sides of the resulted equation by  j ( x, y) , and finally integrating over the nanoplate surface area, the governing PDE converts to a set of ODEs as follows: M(t )q(t )  C(t )q(t )  K(t )q(t )  F(t ),

(45)

17

where the components of vectors and matrices are presented in appendix A. Equation (45) describes transverse vibrations of the nanoplate embedded in a Pasternak foundation and thermal environment while acted by the motion of a moving nanoparticle along an arbitrary path. Defining an elliptical path for the nanoparticle motion on the nanoplate’s surface (as shown in Fig. 2), its time dependent location is defined as follows:

xm  t   x0  a0 cos   t  ,

(46)

ym  t   y0  b0 sin   t  ,

where  is the mass rotation frequency, x0 and y0 are coordinates of the nanoplate center, and a0 and b0 are the large and small radii of the elliptical path. The nonlocal ODE governing the simply supported nanoplate carrying a moving nanoparticle is presented in appendix B.

Fig. 2 A nanoparticle moving on an elliptical path over the nanoplate’s surface 4. Energy-rate method The energy-rate method is a numerical approach to analyze stability characteristics of linear and non-linear systems under parametric excitation. This method based on numerical

18

integration of the accumulated energy of a system in one cycle can be applied in the analysis of periodic systems. It was first introduced by Nakhaei Jazar [25] and then was used only in [33] to analyze stability of Mathieu equation. Unlike perturbation methods which validation of the results rely on the smallness of the book-keeping parameter, the energy-rate method results are valid without any limitation. A brief overview of this approach is presented below. The energy-rate method can be applied to any time-varying second-order ODE as:

x  f  x   g  x, x, x, t   0,

(47)

where f  x  is a single-variable function and g  x, x, x, t  is a time-varying function with the following conditions:

g  0,0,0, t   0, (48)

g  x, x, x, t  T   g  x, x, x, t  .

In addition, functions f and g can be dependent on a limited set of parameters. Eq. (47) may be assumed as a model of a unit mass connected to a spring and subjected to a nonconservative force  g  x, x, x, t  . Assuming the kinetic energy T  x   x , the potential 2 2

energy V  x  

 f  x  dx and the mechanical energy E  T  x   V  x  , the time derivative of

the energy function can be written as follows:

E

d d 1  E    x2  dt dt  2



  f  x  x dx  dt    x . g  x, x, x, t  .

(49)

Now, in order to assess the system stability, the average energy during one period is calculated as follows:

19

T

0

1 1 Eav  E dt  x . g  x, x, x, t  dt , T 0 TT





(50)

Now, if for the selected system parameters, Eav is greater than zero, energy is transferred to the system and, as a result, these parameters are located in an unstable region. But, if Eav is less than zero, the selected parameters belong to a stable region leading to a decrease in the system’s energy. In addition, on the boundaries between stable and unstable regions, Eav is equal to zero and in this case, the selected parameters belong to a transition curve. In this research, the energy-rate method is applied to the parametric equation governing the system to calculate the boundary curves separating stable and unstable regions in the    plane. For this purpose, first the governing equation is rearranged in the form of Eq.

(47). Then, to identify the parameters belonging to the transition curves by the energy-rate method, an algorithm is developed in a computer program so that by meshing the problem parameters plane, it solves the governing differential equation numerically for each pair

 ,  

and calculates Eav , and finally determines the stability boundary curves, i.e.

everywhere Eav

1.

5. Results 5.1 Validation To the best authors’ knowledge, there is no study on the topic of stability analysis and parametric resonance of SLGSs carrying moving nanoparticles. Thus, the results of this study cannot be verified directly with available literature. Hence, neglecting the moving nanoparticle, elastic medium and thermal environment, we try to compare different frequency ratios of SLGSs (ratio of nonlocal natural frequency to the local one) with other studies. Frequency ratio values for different nonlocal parameters for an SLGS with dimensions 20

a  10 nm and b  10 nm are given in Table 1. As can be seen, there is a very good

agreement between the results of this research and those presented by Shen et al. [34], Pradhan and Phadikar [12], and Pradhan and Kumar [35]. Table 1 A comparison between the present work and vibration analysis of SLGSs without nanoparticle

 e0a 

2

nm2

nl

l

nl

l

nl

l

nl

l

Present work

Ref. [34]

Ref.[12]

Ref.[35]

0

1

1

1

1

1

0.9139

0.9139

0.9139

0.9139

2

0.8467

0.8467

0.8467

0.8468

3

0.7925

0.7925

0.7925

0.7926

5.2 Discussion This section presents the stability analysis of transverse vibrations of a SLGS embedded in elastic medium and thermal environment and acted by an orbiting nanoparticle. The energyrate method is applied to examine effects of nonlocal parameter, nanoplate length-to-width ratio, the radii of motion path, temperature change of the thermal environment, stiffness of the Pasternak foundation, and boundary conditions of SLGS on the parametric regions. This study considers an SLGS with length a=20nm, width b=20nm, thickness h  1nm the 3 Young’s modulus E  1.06TPa , density   2250Kg m and the Poisson’s ratio of

  0.25 like what was considered in Ref. [17]. The unstable region in the    plane is completely related to occurrence of principal parametric resonance in the aforementioned system. This type of resonance occurs when the excitation frequency is nearly twice the

21

fundamental natural frequency of the system, which results in unlimited increase of oscillation amplitude. To improve dynamic stability of SLGSs carrying moving nanoparticles in practical applications such as atomic dust detectors, mass sensors and nanoswitches, it is important to study the effect of different parameters on the unstable region. Initially, to investigate the effects of nanoplate length-to-width ratio and the nonlocal parameter on the system stability, the elastic medium and the thermal environment are ignored. Fig. 3 shows stable and unstable regions of the simply supported SLGS in the frequency-mass plane of the moving nanoparticle. The horizontal and vertical axes express the ratio of the orbiting nanoparticle frequency to the fundamental natural frequency of the SLGS and the ratio of the nanoparticle mass to the SLGS mass, respectively. The enclosed area between two curves is the unstable region and other regions are stable. As can be deduced from this figure, the mass of the moving nanoparticle is effective on the stability of the nanostructure vibrations so that when it increases, the range of frequencies for which the system resonates will increase. In addition, by increasing the nanoparticle mass, the critical frequencies decrease and vice versa. In Fig. 4, the effect of SLGS length-to-width ratio, ar , on the unstable region is studied. As can be seen, by increasing the length-to-width ratio, the unstable region shifts to the left, i.e. towards lower frequencies of the orbiting nanoparticle. This is because that increasing the nanoplate length-to-width ratio makes the nanostructure softer and as a result, the parametric resonance frequencies of the system decreases.

22

Fig. 3 The unstable region of the parameters plane; ar  1 ,   1nm2 ,   0.15 ,   0.3 , kw  0 , G p  0 , and T  0 .

Fig. 4 Effect of the SLGS length-to-width ratio on the unstable region;   1nm2 ,   0.15 ,   0.3 , kw  0 , G p  0 , and T  0 .

23

Figure 5 (a)-(d) investigate the effect of nonlocal parameter on the parametric region for different values of the length-to-width ratio including ar  1 , ar  2 , ar  3 and ar  4 , respectively. As can be seen, increasing the value of the nonlocal parameter causes a decrease in the parametric resonance frequencies of the system; that is, increasing the nonlocal parameter leads to a shift of the unstable region toward lower rotating frequencies of the moving nanoparticle. The reason for this can be interpreted as an increase in the nonlocal parameter would reduce the interaction forces between the SLGS atoms and hence makes it softer. Furthermore, trend of variations in Fig. 5 shows that as the ratio of the SLGS lengthto-width increases, the impressibility of the instability trap from the non-local parameter decreases, and for different values of the nonlocal parameter, the corresponding boundaries of different unstable regions tend to become closer. In other words, the small-scale effects influence the stability chart more intensely for lower length-to-width ratio values.

24

(a)

(b)

(c)

(d)

Fig. 5 Effect of the nonlocal parameter on the unstable region;   0.15 ,   0.3 , kw  0 , G p  0 , and T  0 , (a) ar  1 , (b) ar  2 , (c) ar  3 , (d) ar  4 .

In order to investigate the effect of the nanoparticle motion path radii on the nanostructure dynamic stability, Fig. 6 is presented. The results indicate that as the orbiting nanoparticle moves closer to the center of the nanoplate, the area of the unstable region decreases implying the importance of the motion radii on the dynamic stability of the SLGS carrying the moving nanoparticle.

25

Fig. 6 Effect of the nanoparticle motion radii on the unstable region; ar  1 ,   1nm2 , kw  0 , G p  0 , and T  0 .

Figures 7 and 8 investigate the effect of temperature changing of the thermal environment on the stability chart for high temperature



T



T

 1.1106 K -1  and low temperature

 1.6 106 K -1  cases, respectively. For the case of high temperature, increasing the

value of temperature change moves the origin of the unstable region toward lower frequencies of the orbiting nanoparticle due to the resultant compressive force caused by the thermal loading. In fact, for the high temperature case, further increase of the temperature leads to more instability of the dynamical system. On the other hand, for the low temperature case, increasing the value of temperature change augments the parametric resonance frequencies. In this case, because of the negative thermal expansion coefficient, the temperature variation results in a tensional axial force which in turns would amplify rigidity and stability of the nanostructure.

26

Fig. 7 Effect of temperature changes on the unstable region in the case of high temperature;

T  1.1106 K-1 , ar  1 ,   1nm2 ,   0.15 ,   0.3 , kw  0 , and G p  0 .

Fig. 8 Effect of temperature changes on the unstable region in the case of low temperature;

T  1.6 106 K-1 , ar  1 ,   1nm2 ,   0.15 ,   0.3 , kw  0 , and G p  0 .

27

The effects of stiffness of springs and the shear layer of the elastic foundation on the parametric resonance frequencies are examined in Figs. 9 and 10. The results show that increasing the stiffness of the elastic medium improves stability of the nanostructure so that increasing the stiffness of the springs and the shear layer moves the unstable region to higher frequencies; implying more stable system at lower frequencies of the orbiting nanoparticle. In fact, escalation of the foundation stiffness increases the total stiffness of the nanostructure.

Fig. 9 Effect of the foundation springs on the unstable region; T  1.6 106 K-1 , ar  1 ,

  1nm2 , G p  0 , and T  100K .

28

Fig. 10 Effect of the foundation shear layer on the unstable region; T  1.6 106 K-1 , ar  1 ,

  1nm2 , kw  108 N/m3 , and T  100K .

In all abovementioned results, the boundary conditions of the SLGS were considered to be simply supported. Fig. 11 presents the influence of nanoplate boundary conditions on the stability diagram. As shown, the borders of instability region are different for SSSS (all of the edges are simply supported) and CCCC (all of the edges are clamped) boundary conditions. As can be seen, although the width of the unstable region for both boundary conditions seems to be the same, instability occurs for the CCCC plate in higher rotation frequencies of the moving nanoparticle in comparison with the SSSS plate. It can be interpreted that the CCCC boundary conditions represents a stronger constrained case in contrast with the SSSS one. Therefore, the influence of boundary conditions on the parametric resonance of the embedded SLGS carrying moving nanoparticle is noticeable and any negligence in modeling the physical system may lead to unwanted damages in related applications.

29

Fig. 11 Effect of the boundary conditions on the unstable region; T  1.6 106 K-1 , ar  1 ,

  1nm2 , kw  108 N/m3 , G p  2 , and T  100K . 6. Conclusions Investigation of dynamic stability and parametric resonance conditions of SLGSs carrying moving nanoparticles, taking into account inertial effects of the moving nanoparticle and considering elastic medium and thermal environment, was done. Taking into account smallscale effects, the PDE governing the problem was derived based on Kirchhoff’s nonlocal theory and using the Hamilton’s principle. To solve the PDE, the Galerkin method along with trigonometric shape functions was used. Then, the boundaries separating stable and unstable regions in the plane of nanoparticle parameters were calculated using the energy-rate method. The following conclusions were drawn: 

Due to motion of the nanoparticle along an elliptical path, an unstable region appears in the rotation frequency-mass plane of the nanoparticle, which carelessness about that may lead to undesired consequences in practical applications.

30



By increasing the mass of the moving nanoparticle, the range of parametric resonance frequencies becomes wider.



When the nanoparticle mass increases, critical frequencies decrease and vice versa.



By increasing length-to-width ratio of the SLGS, the unstable region moves toward lower rotational frequencies of the moving nanoparticle.



An increase in the value of the nonlocal parameter reduces the parametric resonance frequencies of the system; that is, increasing the nonlocal parameter shifts the unstable region toward lower rotating frequencies of the moving nanoparticle.



As the orbiting nanoparticle path approaches the center of the nanoplate, the area of the unstable region decreases, but the instability region shifts to lower frequencies of the rotating mass.



For high temperature case, increasing the value of temperature change shifts the unstable region origin toward lower frequencies of the orbiting nanoparticle. On the other hand, for the case of low temperature, an increase in the temperature change postpones instability of the system.



Increasing stiffness of the springs and the shear layer of the elastic foundation moves the instability boundaries to higher values of the rotation frequencies of the nanoparticle.



Boundary conditions play an important role on the parametric resonance of the SLGS acted upon by moving nanoparticle which cannot be ignored in practical applications.

31

Appendix A Components of vectors and matrices of Eq. (45).   M ij  i  x ,y   j  x ,y  dx dy    i ,xx  x ,y   j  x ,y  dx dy  i , yy  x ,y   j  x ,y  dx dy    A A A  m m    x , y   x , y    i ,xx  x m , y m   j  x m , y m   i , yy  x m , y m   j  x m , y m   , h i m m j m m h







m  x   x , y    x , y   y m i , y  x m , y m  j  x m , y m    h m i ,x m m j m m m 2   x m i ,xxx  x m , y m   j  x m , y m   x m i ,xyy  x m , y m   j  x m , y m  h

Ci j  2

 y m i ,xxy  x m , y m   j  x m , y m   y m i , yyy  x m , y m   j  x m , y m   , D h

K ij    

NT h

 

 x ,y   j  x ,y   2i ,xxyy  x ,y   j  x ,y   i , yyyy  x ,y   j  x ,y   dx dy

i , xxxx

A

 

i , xx

 x ,y   j  x ,y   i , yy  x ,y   j  x ,y   dx dy

A

NT  h

 

i , xxxx

 x ,y   j  x ,y   2i ,xxyy  x ,y   j  x ,y   i , yyyy  x ,y   j  x ,y   dx dy

A

m x m2 i ,xx  x m , y m   j  x m , y m   y m2 i , yy  x m , y m   j  x m , y m   h

2x m y m i ,xy  x m , y m   j  x m , y m   x m i ,x  x m , y m   j  x m , y m   y m i , y  x m , y m   j  x m , y m   

m   x m2 i ,xxxx  x m , y m   j  x m , y m  h

 x m2 i ,xxyy  x m , y m   j  x m , y m   y m2 i ,xxyy  x m , y m   j  x m , y m   y m2 i , yyyy  x m , y m   j  x m , y m   2x m y m i ,xxxy  x m , y m   j  x m , y m  2x m y m i ,xyyy  x m , y m   j  x m , y m   x m i ,xxx  x m , y m   j  x m , y m   x m i ,xyy  x m , y m   j  x m , y m   y m i ,xxy  x m , y m   j  x m , y m   y m i , yyy  x m , y m   j  x m , y m   



kw   x ,y   j  x ,y  dx dy h i A

  

Fj 

Gp

h

 

i , xx

A

kw  h

 

Gp

 

h

 x ,y   j  x ,y   i , yy  x ,y   j  x ,y   dx dy



i , xx

 x ,y   j  x ,y   i , yy  x ,y   j  x ,y   dx dy

A

i , xxxx

 x ,y   j  x ,y   2i ,xxyy  x ,y   j  x ,y   i , yyyy  x ,y   j  x ,y   dx dy

A

(A.1)

mg   x t  , ym t   . h j m 32

Appendix B Substituting Eq. (43) into Eq. (45), the equation governing the fundamental mode of the system is obtained as: 2 2  m  a   b  2  a b  1    cos 2   0 cos   t   cos 2   0 sin   t     2 2 4  hab  a   b   ab   2 2 m  a   b   a  b   2  cos 2   0 cos   t   cos 2   0 sin   t    2 2   q  hab  a   b   a b 

4

 m  a0  a   a   b   8  sin  t  cos   0 cos   t   sin   0 cos   t   cos 2   0 sin   t   2  a   a   b    ha b 8

2 2 m  3 a0  a0   a0  2  b0  a  b   sin  t cos  cos  t sin  cos  t cos  sin  t               2 2   ha 2b  a   a   b  a b 

8

m  b0  a   b   b   cos  t  cos 2   0 cos  t   cos   0 sin  t   sin   0 sin  t   2  hab  a   b   b 

8

2 2 m  3 b0 a0   b0   b0   a  b  2  cos  t cos  cos  t cos  sin  t sin  sin  t              2 2  q  hab 2  a   b   b   a b  2

 4D  a 2  b 2  m  2a02 2  a   b    sin 2  t  cos 2   0 cos   t   cos 2   0 sin   t    2 2  4 3  ha b  a   b   h  a b  4

2 2 m  4 a02 2 a0  2  b0  a  b  2 2  sin  t cos  cos  t cos  sin  t           2 2   ha 3b  a   b  a b 

4

m  2b02 2  a   b   cos 2  t  cos 2   0 cos   t   cos 2   0 sin   t   3  hab  a   b 

4

2 2 m  4 b02 2 a0  2  b0  a  b  2 2  cos  t cos  cos  t cos  sin  t         2 2   hab 3  a   b  a b 

m  2a0b0 2  a   b  8  sin  t  cos  t  sin   0 cos  t   sin   0 sin  t   2 2  ha b  a   b  m  a0b0 2  a   b  cos   0 cos  t   cos   0 sin  t    8  sin  t  cos  t   ha 2b 2  a   b  4

2 2  a0   b0   a0   b0  a  b  sin   cos  t   sin   sin  t   cos   cos  t   cos   sin  t    2 2   a   b   a   b  a b 

4

m  a0 2  a   a   b   cos  t  cos   0 cos  t   sin   0 cos  t   cos 2   0 sin  t   2  ha b  a   a   b 

33

4

2 2 m  3 a0 2  a0   a0  2  b0  a  b   cos  t cos  cos  t sin  cos  t cos  sin  t            2 2   ha 2b  a   a   b  a b 

4

m  b0 2  a   b   b   sin  t  cos 2   0 cos  t   cos   0 sin  t   sin   0 sin  t   2  hab  a   b   b 

2 2 m  3 b0 2 a0   b0   b0  a  b  2 4  sin  t  cos   cos  t   cos   sin  t   sin   sin  t    2 2   hab 2  a   b   b  a b 

N T  2  a2  b 2  N T  4  a2  b 2  kw kw  2  a2  b 2  G p  a2  b 2              h  a 2b 2   h  a 2b 2   h  h  a 2b 2   h  a 2b 2  2

2

2 G p 4   a 2  b 2   mg  a   b   cos   0 cos  t   cos   0 sin  t  .  2 2  q  2  h  a b   ab  h  a   b 

(B.1)

In order to generalize the extracted equation, non-dimensional parameters are defined as follows:



m ,  hab

Q

q , ab

G p*



N T*

 , D  a2  b 2  2     h  a 2b 2  NT , 2 2 2  a  b  2 D ab  2 2   ab 

Gp  a2  b 2   2 D ab  2 2   ab 

2

,



*

ab

,

kw

k w*

 a2  b 2  2 2   ab 

 4D  g

g*

4

a , b

ar

D  a2  b 2  ab    h  a 2b 2 

2

2



t ,

,



a0 , a



b0 b (B.2)

.

Using these non-dimensional parameters besides the chain rule, the dimensionless equation of motion is extracted as follows:

34

  1  ar2  2 2  2 1   2  *    4 cos  cos    cos  sin     ar    1  ar2    4  cos  cos    cos  sin       Q a  r  2

*

2

2

   2  8 sin   cos  cos    sin  cos    cos 2  sin      1  ar2   8 3  * sin   cos  cos    sin  cos    cos 2  sin       ar   8 cos   cos 2  cos    cos  sin    sin  sin     1  ar2    8   cos   cos  cos    cos  sin    sin  sin       Q a  r  3

*

2

   1  4 2 2  2 sin 2   cos 2  cos    cos 2  sin      1  ar2   4 4  * 2  2 sin 2   cos 2  cos    cos 2  sin       ar   4 2 2  2 cos 2   cos 2  cos    cos 2  sin     1  ar2   4 4  * 2  2 cos 2   cos 2  cos    cos 2  sin       ar   8 2  2 sin   cos   sin  cos    sin  sin    cos  cos    cos  sin     8 4  *  2 sin   cos   sin  cos    sin  sin    cos  cos    cos  sin     1  ar2  2 2    4  cos   cos  cos    sin  cos    cos  sin    a  r   1  ar2   4    cos   cos  cos    sin  cos    cos  sin       ar  3

*

2

2

 4  2 sin   cos 2  cos    cos  sin    sin  sin     1  ar2   4 3  *  2 sin   cos 2  cos    cos  sin    sin  sin       ar  2 2 2 2 2 2  1  ar2  * * 2  1  ar  * * * 2  1  ar  *  1  ar  * * 2  1  ar  N    NT      kw  kw      Gp    Gp     Q  ar   ar   ar   ar   ar   * T

 2 g * cos  cos    cos  sin    ,

(B.3)

35

where here the prime sign denotes derivation with respect to the dimensionless time variable

 . As can be seen, Eq. (B.3) is an ODE with periodic coefficients (with periodicity of 2 ) stating a parametric-type excitation which may result in dynamic instability of the system. So, in the following, the energy-rate method is used to find those values of mass and rotation frequency of the moving nanoparticle which induce instability in the plate vibrations.

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