Transient behavior of an orthotropic graphene sheet resting on orthotropic visco-Pasternak foundation

International Journal of Engineering Science 103 (2016) 97–113

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Transient behavior of an orthotropic graphene sheet resting on orthotropic visco-Pasternak foundation A. Ghorbanpour Arani a,b,∗, M.H. Jalaei a a b

Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran Institute of Nanoscience& Nanotechnology, University of Kashan, Kashan, Iran

a r t i c l e

i n f o

Article history: Received 18 September 2015 Revised 3 January 2016 Accepted 7 February 2016 Available online 20 April 2016 Keywords: Transient analysis Viscoelastic orthotropic graphene sheet Fourier series-Laplace transform Nonlocal elasticity theory Orthotropic visco-Pasternak foundation

a b s t r a c t This paper deals with transient analysis of simply-supported orthotropic single-layered graphene sheet (SLGS) resting on orthotropic visco-Pasternak foundation subjected to dynamic loads. The size effect is taken into account using Eringen’s nonlocal theory due to its simplicity and accuracy. In order to present a realistic model, the material properties of graphene sheet are supposed viscoelastic using Kelvin–Voigt model. Based on the first order shear deformation theory (FSDT), equations of motion are derived using Hamilton’s principle which are then solved analytically by means of Fourier series-Laplace transform method. The present results are found to be in good agreement with those available in the literature. Some numerical results are presented to indicate the influences of size effect, elastic foundation type, structural damping, orthotropy directions and damping coefficient of the foundation, modulus ratio, length to thickness ratio and aspect ratio on the dynamic behavior of rectangular SLGS. Results depict that the structural and foundation damping coefficients are effective parameters on the transient response, particularly for large damping coefficients, where response of SLGS is damped rapidly. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Graphene, a single atomic layer of graphite arranged in a two dimensional honeycomb structure, is one of the most famous and beloved types of carbon structures among researchers because of its superior electrical, thermal, chemical, optical and mechanical properties. For this reason it is vastly used in nano-electro-mechanical systems (NEMS) such as sensors (Murmu & Adhikari, 2013; Sakhaee-Pour, Ahmadian, & Vafai, 2008), nano-sheet resonators (Eichler et al., 2011), nanoactuators (Ji et al., 2012), conductive electrodes for solar cells (Wang, Zhi, & Müllen, 2008) and so on. Hence, investigating the behavior of nano-mechanical systems made of graphene helps in better designing. Generally, three major procedures have been developed to analyze the mechanical properties of nanostructures known as molecular dynamics (MD) simulations, experimental study and continuum mechanics approach. The first two methods are very cumbersome and computationally prohibitive for nanostructure systems with large number of atoms. Thus, because of these limitations, continuum mechanic approaches have been known as powerful and effective methods to study mechanical characteristics of nanostructures. Since the classical continuum mechanics have no ability in capturing the small scale effects, it cannot be regarded as a reliable theory to predict the mechanical behavior of nanomaterials. So far, several nonclassical continuum theories have been formulated to incorporate the small-scale size effects in micro/nano structures, such ∗

Corresponding author at: Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran. Tel.: +98 31 55912450; fax: +98 31 55912424. E-mail address: [email protected], [email protected] (A.G. Arani).

http://dx.doi.org/10.1016/j.ijengsci.2016.02.006 0020-7225/© 2016 Elsevier Ltd. All rights reserved.

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as nonlocal elasticity theory (Eringen, 1972; Lei, Adhikari, & Friswell, 2013; Najar, El-Borgi, Reddy, & Mrabet, 2015; Reddy, 2007; Reddy & El-Borgi, 2014), strain gradient theory (Ghayesh, Amabili, & Farokhi, 2013; Kong, Zhou, Nie, & Wang, 2009; Lam, Yang, Chong, Wang, & Tong, 2003; Wang, 2010) and couple stress theory (Akgöz & Civalek, 2012; Dai, Wang, & Wang, 2015; Ghayesh & Farokhi, 2015; Ghorbanpour Arani, Abdollahian, & Jalaei, 2015; Mohammad-Abadi & Daneshmehr, 2014). Among these size dependent theories, the nonlocal elasticity theory initiated by Eringen (1983, 2002) has been widely used in the study of structures at small scale. In this theory, the nonlocal stress tensor at a reference point in a body depends not only on the strain tensor at that point, but also on the strain tensor at all other points in the body. The literature shows that nonlocal theory is being increasingly utilized for reliable and quick analysis of nanostructures in recent years. In this regard, a number of research works have been performed based on this theory in order to study on the bending (Aghababaei & Reddy, 2009; Scarpa, Adhikari, Gil, & Remillat, 2010; Wang & Li, 2012), buckling (Ansari & Rouhi, 2012; Daneshmehr, Rajabpoor, & Pourdavood, 2014; Karamooz Ravari, Talebi, & Shahidi, 2014; Narendar, 2011; Samaei, Abbasion & Mirsayar, 2011; Sarrami-Foroushani & Azhari, 2014) and vibration (Aghababaei & Reddy, 2009; Assadi, Farshi, & Alinia-ZiaZi, 2010; Daneshmehr, Rajabpoor, & Hadi, 2015; Jomehzadeh & Saidi, 2011; Liew, He, & Kitipornchai, 2006; Malekzadeh & Shojaee, 2013; Pradhan & Kumar, 2011; Pradhan & Phadikar, 2009; Sarrami-Foroushani & Azhari, 2014) of graphene sheets. Reddy, Rajendran, and Liew (2006) computed elastic constants of the graphene sheet based on the Brenner’s potential and the CauchyBorn rule. It has been shown that due to the variation in coordination number, the equilibrium bond length of carbon-carbon is not uniform everywhere in the graphene. They found that graphene behaves like an orthotropic material. Mohammadi, Farajpour, Goodarzi, and Shehni nezhad pour (2014) presented nonlocal theory to study the free vibration of orthotropic single-layered graphene sheet (SLGS) resting on a Pasternak foundation under shear in-plane load based on classical plate theory (CLPT) and used the combined Galerkin-differential quadrature method to solve the obtained equations. They concluded that increasing nonlocal parameter reduces the non-dimensional frequency of the SLGS. Also, the small scale effects are more significant for the nanoplate with shear in-plane load compared to nanoplate without shear in-plane load. Arash, Wang, and Liew (2012) investigated an inclusive research on wave propagations in SLGS by the nonlocal finite element plate model and MD simulations. They found that nonlocal finite element (FE) model is essential in analysis of graphene sheets (GSs), especially at wavelengths less than 1 nm. The elastic buckling and vibration analyses of isotropic and orthotropic GSs under biaxial compression and pure shear loading based on Eringen’s nonlocal theory using the spline finite strip method were reported by Analooei, Azhari, and Heidarpour (2013). They revealed that the buckling behavior of nanoplate subjected to shear loading is more sensitive to the small scale effects than it under biaxial loading. Also, their work indicated that by increasing the dimensions of nanoplate, size effect reduces. Ansari, Rajabiehfard, and Arash (2010) reported a finite element method (FEM) based on the nonlocal theory to investigate the small scale effect on the vibration analysis of multi-layered graphene sheets (MLGSs) with various boundary conditions embedded in an elastic medium. They found that by increasing nonlocal parameter, the size dependency increase in all of boundary conditions. In addition, their results indicated that the natural frequencies more sensitive to the nonlocal parameter in higher mode number. Rayleigh-Ritz solution for nonlocal vibration behavior of isotropic rectangular nanoplates with different boundary conditions on the basis of CLPT was presented by Chakraverty and Behera (2014). They observed that when the aspect ratio increases, the frequency parameter increases. Their work also showed that frequency parameters are highest in nanoplate with fully clamped boundary condition. Nonlinear nonlocal vibration response of the coupled system of double-layered annular graphene sheets (CS-DLAGSs) embedded in a visco-Pasternak medium was investigated numerically using differential quadrature method (DQM) by Ghorbanpour Arani, Maboudi, and Kolahchi (2014). They revealed that the frequency reduction percent (FRP) of in phase-in phase-in phase (III) and out phase-out phase-out phase (OOO) vibration state are maximum and minimum, respectively. In addition, their results indicated that the FRP of Visco-Winkler and Pasternak mediums are maximum and minimum, respectively. Employing nonlocal theory and von-Kármán model, Naderi and Saidi (2014) researched postbuckling behavior of orthotropic GSs in nonlinear polymer medium under both uniaxial and biaxial in-plane loadings. They solved equilibrium equations via the Galerkin method for GSs with various boundary conditions based on the CLPT. They observed that the small scale effects are obvious especially on Postbuckling behavior of nanoplate having stiffer boundary conditions. Also, their work showed that when the external loads increases, the nonlinearity effect increases. All the above mentioned researches have been conducted on the nonlocal continuum models for buckling and free vibration of graphene sheets, however a little attention has been devoted to the bending problem of graphene sheets based on the nonlocal elasticity theory. In this regard, Kanaipour (2014) studied static bending analysis of nanoplate embedded on elastic foundation. The governing equations for the nonlocal Mindlin and Kirchhoff plate models were derived and then were solved numerically using DQM. He revealed that when the nanoplate becomes thicker, nonlocal Mindlin plate model is more appropriate. Also, he observed that by increasing the elastic stiffness, the displacement ratio increases. The nonlinear bending response of rectangular orthotropic SLGS resting on Pasternak foundation, subjected to uniform load presented by Golmakani and Rezatalab (2014) based on nonlocal first order shear deformation theory (FSDT). The governing equations were obtained with assumption of von-Kármán relationship and then were solved using DQM for various types of boundary conditions. Their results showed that when the elastic foundation exists, the linear to nonlinear deflection ratio decreases. In the field of transient analysis of nanoplate, Liu and Chen (2014) analyzed dynamic response of the finite periodic SLGSs with different boundary conditions using the wave method on the basis of the nonlocal Mindlin plate theory. They found that dynamic displacement responses of finite GSs can be reduced much by periodic arrangement design. Also, their work indicated that the transverse shear strain responses for the periodic nanoplate in the band gap frequency domain are much smaller than those of uniform ones. Most recently, Ghorbanpour Arani and Jalaei (2015) investigated static bending

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and dynamic behavior of embedded isotropic elastic SLGS based on nonlocal third-order shear deformation theory. The surrounding medium was simulated by isotropic visco-Pasternak model. They revealed that when the nonlocal parameter increases, the dynamic response increases. Also, their results indicated that the small scale effect on the deflection is more prominent as the nanoplate becomes thicker and the aspect ratios increases. None of the research works which are mentioned above have modeled nanoplates as viscoelastic structures whereas the nanoplates reveal viscoelastic structural damping as many materials. Recently, vibration analysis of a simply supported viscoelastic orthotropic nanoplate resting on viscoelastic foundation was performed by Pouresmaeeli, Ghavanloo, and Fazelzadeh (2013) who showed that increasing the structural damping and foundation damping coefficients diminishes the ´ Kozic, ´ and Pavlovic´ (2014) carried out the free vibration of a viscoelastic orfrequency of orthotropic nanoplates. Karlicˇ ic, thotropic multi-nanoplate system (MNPS) by including the effect of viscoelastic foundation, based on the Kirchhoff plate theory and assumed Navier solutions. They found that increasing the number of nanoplates causes to increase the damped natural frequency and also decreases the damping ratio. Wang, Li, and Wang (2015) presented analytical solutions for nonlinear vibration of the double layered viscoelastic nanoplates with simply supported and clamped boundary conditions based on CLPT with assumption of von-Kármán strains using the method of multiple scales. Their results indicated that the vdW interaction has considerable effects on the natural frequency, while the effect of the structural damping coefficient on the nonlinearity frequency is not significant. However, to the best of authors’ knowledge, no study has focused on transient analysis of viscoelastic orthotropic SLGS resting on orthotropic visco-Pasternak foundation under sinusoidal and uniform dynamic loads so far. Considering viscoelastic characterization both GS and foundation are very significant for perfect analysis of NEMS. Hence, in this work an entirely analytical method to study the dynamic responses of SLGS with considering foundation effects is developed using the Fourier series-Laplace transform. The material properties of GS are assumed to be orthotropic and viscoelastic. Viscoelasticity of the structure material is modeled with parallel springs and dashpots as the Kelvin–Voigt model. The SLGS resting on the viscoelastic medium is simulated by orthotropic visco-Pasternak type as spring, shear and damping foundations with considering shear direction and orthotropy angle. Based on the FSDT, equations of motion are derived employing Hamilton’s principle. Furthermore, the nonlocal elasticity theory is applied to capture the small scale effects. Using Laplace transform, the time dependency of the governing equations is eliminated and then an analytical strategy is employed to invert the results into the time domain. Finally, the influences of small scale effect, modulus ratio, elastic foundation type, structural damping, orthotropy directions and damping coefficient of the foundation, loading type, length to thickness ratio and aspect ratio on the dynamic behavior of SLGS are discussed in detail. The obtained results would be helpful while designing NEMS devices using GSs.

2. Basic equations A schematic configuration of the viscoelastic orthotropic SLGS with length a, width b and thickness h resting on an orthotropic visco-Pasternak foundation subjected to dynamic transverse uniform and sinusoidal loads has been illustrated in Fig. 1. The viscoelastic nanoplate is described based on the Kelvin–Voigt model consists of an infinite set of springs and dashpots in parallel. This model handles the characteristics of creep and recovery fairly well. As shown, due to the presence of arbitrarily orthotropic foundation, the global coordinates of the nanoplate (x, y, z) will not coincide with the local orthotropy coordinates (ξ , η) of the medium. 2.1. Nonlocal continuum theory The conventional local theory is not size dependent theory. Thus, it is required to modify this theory to include small scale effects. For this purpose, nonlocal elasticity theory was suggested by Eringen (1983). Due to its efficiency and simplicity, it has been extensively applied. According to the nonlocal theory, the stress tensor at an arbitrary point in a body depends not only on the strain tensor at that point but also on the strain tensor at all other points of the body. This observation is in accordance with atomic theory of lattice dynamics and experimental observations on phonon dispersion. Using nonlocal elasticity theory, the constitutive equation for a linear homogenous nonlocal elastic body neglecting the body forces is given as:

σi j nl (x ) =

 v

  α (x − x , τ ) σi j l dV (x ),

∀x ∈ V

(1)

where σinlj and σilj are the nonlocal and local stress tensors, respectively. The term α (|x − x |, τ ) is the nonlocal modulus, which incorporates nonlocal effects into the constitutive equation at the reference point x produced by the local strain at the source x ; |x − x | represents the distance between x and x in the Euclidean form, and τ = e0 a/l in which l is the external characteristic length (e.g., crack length, wavelength), a is an internal characteristic length of the material (e.g., length of C–C bond, lattice parameter, granular distance), and e0 indicates constant appropriate to each material, and consequently, e0 a is a constant parameter which is obtained with the experimental observations or MD simulation results. It should be noted that when e0 a is equal to zero, the nonlocal elasticity reduces to the local (classical) elastic model. The differential form of

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Fig. 1. (a) Coordinate system and geometric of viscoelastic orthotropic SLGS embedded in orthotropic visco-Pasternak foundation; and (b) schematic of viscoelastic orthotropic SLGS under uniform and sinusoidal loads.

Eq. (1) can be written as:

(1 − μ ∇ 2 )σ nl = C : ε

(2)

In the above equation, the parameter μ = (e0 denotes the small scale effect on the response of structures in nanosize 2 2 and ∇ 2 = ∂∂x2 + ∂∂y2 is the Laplacian operator in a Cartesian coordinate system. Also, C is the fourth order stiffness tensor, ‘:’ a)2

represents the double dot product and ε is the strain tensor. Using Eq. (2), the constitutive equation of the orthotropic nanoplate can be expressed as:

⎧ nl ⎫ ⎧ nl ⎫ ⎡ σxx ⎪ σxx ⎪ C11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨σyynl ⎪ ⎬ ⎨σyynl ⎪ ⎬ ⎢C21 σyznl − μ ∇ 2 σyznl = ⎢ ⎣0 ⎪ ⎪ nl ⎪ nl ⎪ ⎪ ⎪ ⎪ ⎪ 0 σ σ ⎪ ⎪ ⎪ ⎪ xz xz ⎩ nl ⎭ ⎩ nl ⎭ 0 σxy σxy

C12 C22 0 0 0

0 0 C44 0 0

0 0 0 C55 0

⎤⎧

⎫

0 ⎪ εxx ⎪ ⎨εyy ⎪ ⎬ 0 ⎥⎪ ⎥ 0 γ yz . ⎦ ⎪ ⎪ 0 ⎪ ⎩γxz ⎪ ⎭ C66 γxy

(3)

The coefficients of Cij are the plane stress-reduced stiffness of the orthotropic nanoplate defined as follows (Reddy, 2004):

C11 =

E1 , 1 − ν12 ν21

C12 =

ν12 E2 , 1 − ν12 ν21

C22 =

E2 , 1 − ν12 ν21

C66 = G23 ,

C44 = G13 ,

C55 = G12

(4)

where E1 and E2 are Young’s moduli in directions x and y, respectively. G12 , G13 and G23 denote the shear moduli and ν 12 and ν 21 are Poisson’s ratios. 2.2. Strain displacement relationships In this study, to capture the thickness shear deformations and rotary effects, the FSDT is utilized to formulate the governing equations. Based on the FSDT (Reddy, 2004), the mid-surface displacements (u0 , v0 , w0 ), mid-surface rotations (φ x , φ y ) and the displacement components of an arbitrary point (u, v, w) are in association as:

u(x, y, z, t ) = u0 (x, y, t ) + zφx (x, y, t ), v(x, y, z, t ) = v0 (x, y, t ) + zφy (x, y, t ), w(x, y, z, t ) = w0 (x, y, t ),

(5)

in which t denotes the time variable. The linear in-plane and transverse shear strains are given by:

 ⎧ ε (0 ) ⎫ ⎧ ε (1 ) ⎫     ⎨ xx ⎬ ⎨ xx ⎬ εxx γyz γyz(0) 0) 1) ( ( εyy = εyy + z εyy , = , γxz ⎩ (0 ) ⎭ ⎩ (1 ) ⎭ γxz(0) γxy γxy γxy



(6)

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where

⎧ (0 ) ⎫ ⎧ ⎫ ⎧ (1 ) ⎫ ⎧ ⎫ ⎨ εxx ⎬ ⎨ u0,x ⎬ ⎨ εxx ⎬ ⎨ φx,x ⎬ v0,y φy,y , , ε (0 ) = ε (1 ) = ⎩ yy(0) ⎭ ⎩ ⎭ ⎩ yy(1) ⎭ ⎩ ⎭ u + v φ + φ x,y y,x 0 ,y 0 ,x γxy γxy

 (0 )    γyz φy + w0,y = . φx + w0,x γxz(0)

101

(7)

Here, the comma in subscript represents the partial differentiation. 3. Energy method 3.1. Strain energy The strain energy of the rectangular SLGS can be written as:

U=

1 2





h 2

− 2h

b a 0

0

(σxx εxx + σyy εyy + Ks σyz γyz + Ks σxz γxz + σxy γxy )dx dy dz

(8)

In the above equation, Ks is the shear correction factor of FSDT. As widely accepted, the approximate value of this quantity is Ks = 5/6 (Reddy, 2004, 2007). 3.2. Kinetic energy The kinetic energy of the SLGS can be obtained as:

K=

ρh 2



b a 0



0

 (u˙ )2 + (v˙ )2 + (w˙ )2 dx dy

(9)

where ρ is the density of the orthotropic graphene sheet and dot-superscript convention shows the differentiation with respect to the time. 3.3. External works The graphene sheet is subjected to the external applied loads and resting on an orthotropic visco-Pasternak elastic foundation. Hence, the external works can be divided to the following two distinct forces: • orthotropic visco-Pasternak medium; • external applied loads. 3.3.1. Elastic medium Winkler foundation or one-parameter model is the simplest simulation of a foundation that considers just the normal stresses. Pasternak foundation or two-parameter model considers not only the normal stresses, but also the transverse shear deformation. This model assumes a shear layer on the top of the Winkler foundation (Ghorbanpour Arani et al., 2015; Khajeansari, Baradaran, & Yvonnet, 2012; Samaei, Abbasion, & Mirsayar, 2011). Taking the advantage of Pasternak’s model, foundation can be defined generally as arbitrary orthotropy directions. Orthotropic visco-Pasternak foundation is simulated by adding damping to the orthotropic Pasternak model. Since the damping coefficient has remarkable effect on the dynamic response of material, it should be considered in the dynamic analysis. Therefore, visco foundation can yield the accurate results with respect to non-visco ones. In this paper, the bottom surface of SLGS is continuously in contact with an orthotropic visco-Pasternak foundation. The force induced by orthotropic visco-Pasternak foundation can be obtained as (Ghorbanpour Arani, Shiravand, Rahi, & Kolahchi, 2012; Kutlu, Ug˘ urlu, Omurtag, & Ergin, 2012): 2 F1 = kw w + cd w˙ − kgξ (cos2 θ w,xx + 2 cos θ sin θ w,xy + sin θ w,yy )

− kgη (sin

2

θ w,xx − 2 sin θ cos θ w,xy + cos2 θ w,yy )

(10)

in which kw , cd , kgξ and kgη are spring, damper, ξ -shear and η-shear constants, respectively. The angle θ describes the local ξ direction of orthotropic foundation with respect to the global x-axis of the nanoplate. 3.3.2. External applied loads The force due to external applied loads can be written as:

F2 = −p

(11)

where p is the intensity of the distributed transverse load. Therefore, the work done due to elastic medium and external forces on the SLGS is

V =−

1 2



b a

0

0

(F1 + F2 )wdx dy

(12)

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4. Equations of motion Applying Hamilton’s principle the variational form of motion equations can be expressed as follows:

δ

 0

t

[K − (V + U )]dt = 0

(13)

Equating the coefficients of δ u0 , tained:

δ v0 ,

δ w0 ,

δφ x and δφ y to zero, the following equations of motion can be ob-

δ u0 : Nxx,x + Nxy,y = I0 u¨ 0 + I1 φ¨ x ,

(14a)

δv0 : Nxy,x + Nyy,y = I0 v¨ 0 + I1 φ¨ y ,

(14b)

δ w0 : Qxz,x + Qyz,y + p(x, y, t ) − kw w0 + kgξ (cos2 θ w0,xx + 2 cos θ sin θ w0,xy + sin2 θ w0,yy ) + kgη (sin

where

2

θ w0,xx − 2 sin θ cos θ w0,xy + cos2 θ w0,yy ) = I0 w¨ 0 + cd w˙ 0 ,

(14c)

δφx : Mxx,x + Mxy,y − Qxz = I1 u¨ 0 + I2 φ¨ x ,

(14d)

δφy : Mxy,x + Myy,y − Qyz = I1 v¨ 0 + I2 φ¨ y ,

(14e)



Nαβ Mαβ



 =

− 2h



h 2

Qα z = Ks Ii = ρ

h 2

−h/2

1 dz z

α = x, y β = x, y

(15a)

σαnlz dz α = x, y

(15b)

zi dz (i = 0, 1, 2 ),

(15c)

− 2h h/2



  nl σαβ

in which Nαβ , Mαβ and Qα z are in-plane, moment and transverse shear stress resultants of nonlocal elasticity, respectively. The stress resultants are related to the strains as follows:

        {N} − μ∇ 2 {N} = [A] [0] ε (0)  , {M } {M } [0 ] [D ] ε (1 )        (0)  γ {Qyz } − μ∇ 2 {Qyz } = J44 0  yz(0)  , 0 J55 {Qxz } {Qxz } γxz

where

 T {N} = Nxx Nyy Nxy , 

  (0 ) (0 ) T ε (0) = εxx εyy γxy(0) ,

Here Aij ,



 T {M} = Mxx Myy Mxy   (1 ) (1 ) T ε (1) = εxx εyy γxy(1)

(16b)

(17a) (17b)

Dij and Jii which are the extensional, bending and shear stiffness of the graphene sheet defined as:





Ai j , Di j = 

Jii = Ks



(16a)

h/2

−h/2

h/2

−h/2





(i, j = 1, 2, 6 ),

Ci j 1, z2 dz

Cii dz = Ks h Cii

( i = 4, 5 )

(18a)

(18b)

Kelvin–Voigt model is employed for considering viscoelastic behavior of the nanostructure. According to this model, Young’s moduli Ei and shear moduli Gij are as follows (Pouresmaeeli et al., 2013):



Ei = Ei

1+g

 Gi j = Gi j

∂ ∂t

1+g



∂ ∂t

( i = 1, 2 ),

(19a)

 ( i = j = 1, 2, 3 )

in which g is the structural damping coefficient.

(19b)

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Substituting Eqs. (15), (18) and (19) into the equations of motion (i.e., Eq. ( 14 )), the governing equations of viscoelastic orthotropic SLGS in terms of displacements and rotations can be obtained as:

A¯ 1 (u0,xx + gu˙ 0,xx ) + A¯ 2 (u0,yy + gu˙ 0,yy ) + A¯ 3 (v0,xy + gv˙ 0,xy ) + (1 − μ∇ 2 )(A¯ 4 u¨ 0 + A¯ 5 φ¨ x ) = 0,

(20a)

B¯ 1 (u0,xy + gu˙ 0,xy ) + B¯ 2 (v0,xx + gv˙ 0,xx ) + B¯ 3 (v0,yy + gv˙ 0,yy ) + (1 − μ∇ 2 )(B¯ 4 v¨ 0 + B¯ 5 φ¨ y ) = 0,

(20b)

C¯1 (w0,xx + gw˙ 0,xx ) + C¯2 (w0,yy + gw˙ 0,yy ) + C¯3 (φx,x + gφ˙ x,x ) + C¯4 (φy,y + gφ˙ y,y )





2 C¯ w¨ − kw w0 + kgξ (cos2 θ w0,xx + 2 cos θ sin θ w0,xy + sin θ w0,yy ) + (1 − μ∇ ) 5 0 = 0, 2 2 +kgη (sin θ w0,xx − 2 sin θ cos θ w0,xy + cos θ w0,yy ) − cd w˙ 0 + p(x, y, t ) D¯ 1 (w0,x + gw˙ 0,x ) + D¯ 2 (φx,xx + gφ˙ x,xx ) + D¯ 3 (φx,yy + gφ˙ x,yy ) + D¯ 4 (φx + gφ˙ x ) + D¯ 5 (φy,xy + gφ˙ y,xy ) 2

+ (1 − μ∇ 2 )(D¯ 6 u¨ 0 + D¯ 7 φ¨ x ) = 0,

(20c)

(20d)

E¯1 (w0,y + gw˙ 0,y ) + E¯2 (φx,xy + gφ˙ x,xy ) + E¯3 (φy,xx + gφ˙ y,xx ) + E¯4 (φy,yy + gφ˙ y,yy ) + E¯5 (φy + gφ˙ y ) + (1 − μ∇ 2 )(E¯6 v¨ 0 + E¯7 φ¨ y ) = 0,

(20e)

where the coefficients A¯ i , B¯ i , C¯i , D¯ i , and E¯i , are given in Appendix A. Considering that the rectangular graphene sheet has simply supported boundary conditions at all four edges, we can write following form (Reddy, 2004):

u0 (x, 0, t ) = 0,

φx (x, 0, t ) = 0,

u0 (x, b, t ) = 0,

φx (x, b, t ) = 0,

(21a)

v0 (0, y, t ) = 0,

φy (0, y, t ) = 0,

v0 (a, y, t ) = 0,

φy (a, y, t ) = 0,

(21b)

w0 (x, 0, t ) = 0,

w0 (x, b, t ) = 0,

w0 (0, y, t ) = 0,

Nxx (0, y, t ) = 0,

Nxx (a, y, t ) = 0,

Nyy (x, 0, t ) = 0,

Mxx (0, y, t ) = 0,

Mxx (a, y, t ) = 0,

Myy (x, 0, t ) = 0,

w0 (a, y, t ) = 0, Nyy (x, b, t ) = 0, Myy (x, b, t ) = 0,

(21c) (21d) (21e)

5. Solution procedure 5.1. Space solution A closed-form Navier’s type solution is employed to solve governing equations. On the basis of Navier solution, the generalized displacements are expanded in a double Fourier series as product of undetermined coefficients and known trigonometric functions to satisfy boundary conditions, i.e., Eq. (21). Hence, the appropriate displacement components can be defined as: ∞  ∞ 

u0 (x, y, t ) =

Umn (t ) cos(α x ) sin(β y ),

(22a)

Vmn (t ) sin (α x ) cos (β y ),

(22b)

Wmn (t ) sin (α x ) sin (β y ),

(22c)

m=1 n=1 ∞  ∞ 

v0 (x, y, t ) =

m=1 n=1 ∞  ∞ 

w0 (x, y, t ) =

φx (x, y, t ) = φy (x, y, t ) =

m=1 n=1 ∞  ∞  m=1 n=1 ∞  ∞ 

Xmn (t ) cos (α x ) sin (β y ),

(22d)

Ymn (t ) sin (α x ) cos (β y ),

(22e)

m=1 n=1

where α = maπ , β = nbπ , and m, n are the half wave numbers in the x and y directions, respectively. Furthermore, as mentioned, it is assumed that the nanoplate is subjected to transverse mechanical load which can be expressed as the following Fourier sin expansion:

p(x, y, t ) =

∞  ∞  m=1 n=1

pmn (t ) sin (α x ) sin (β y )

(23)

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Here, the coefficients pmn (t) for two types of dynamic load distribution at the top surface of viscoelastic orthotropic SLGS are presented as:

16P (t ) , (m, n = 1, 3, 5, ... ) for uniform load mnπ 2 pmn (t ) = P (t ), (m = n = 1 ) for sinusoidal load pmn (t ) =

(24)

where P (t ) = P0 H (t ) , and P0 represents the intensity of distributed applied load. Also, H(t) is the Heaviside step function 1, t≥0 defined as H (t ) = { . 0, t<0 Substituting Eqs. (22) and (23) into the governing Eq. (20), the following system of equations is obtained in a matrix form as:









¨ mn + [Cmn ]  ˙ mn + [Kmn ]{mn } = {Fmn }, [Mmn ] 

(25)

in which {mn } = {Umn Vmn Wmn Xmn Ymn }T is the displacement vector. Furthermore, the [M], [C] and [K] are the mass, damping and stiffness matrices, respectively, which are defined in Appendix B. 5.2. Dynamic response solutions For the dynamic bending analysis, Eq. (25) must be solved by using the Laplace transformation. Performing the Laplace ˙ mn } = 0 at t = 0), yields transform on Eq. (25) and considering zero initial conditions at the initial time (namely, {mn } = { a new system of equations in which time dependency is eliminated as follows:



Kmn + sCmn + s2 Mmn



   ¯ mn = F¯mn , 

(26)

Here, s is the Laplace transform parameter and the bar superscript indicates transformed quantities. Solving the system of Eq. (26), each component of the displacement vector is derived in the Laplace domain. 5.3. Analytical Laplace inversion At the end of previous section, each of the five components of the displacement vector was obtained in the Laplace domain. In this section, an analytical Laplace inversion technique is employed to return the displacement vector from Laplace domain into the real time domain. A function fˆ(s ) = A(s ) can be used to find the unknown variables in Eq. (26) in the Laplace transformation domain. Both B (s )

functions A(s) and B(s) are in the form of polynomials. Let’s suppose that the roots of the function B(s) are known. Some of the roots are real roots which are denoted by ri , and the others are complex and indicated by ci . Number of ri and ci are shown as nr and nc , respectively. When all roots are simple, the inverse of the function fˆ(s ), that is f(t), is obtained as (Kiani, Sadighi, & Eslami, 2013; Krylov & Skoblya, 1977): nc  A ( ci ) ci t f (t ) = Re e B ( ci ) i=1

!

+

nr  A (r j ) r j t e . B ( r j )

(27)

j=1

where Re(x) denotes the real part of the complex number x and the prime specifies a derivative with respect to s . Following the mentioned approach, each component of the displacement vector is derived analytically. 6. Numerical results and discussion In this study, dynamic response of viscoelastic orthotropic SLGS resting on orthotropic visco-Pasternak foundation is carried out. The effects of various parameters such as small scale parameter, structural damping, viscoelastic foundation, kind of the applied load, length to thickness ratio (a/h) and aspect ratio (a/b) on size dependent dynamic of rectangular SLGS are presented graphically and discussed in detail. Since the successful application of the nonlocal continuum mechanics requires to determine the magnitude of the small scale parameter e0 a, an appropriate choice of this parameter had to be made. In the most studies e0 is usually taken to be 0.39 as proposed by Eringen (1972, 1983, 2002). Literatures show that the magnitudes of e0 extremely depend on various parameters, and its actual value is not known so far. Some researchers assumed a range of values e0 a = 0–2 nm for different analyses of GS (Ansari & Rouhi, 2012; Ghorbanpour Arani, Shiravand, Rahi & Kolahchi, 2012; Mohammadi et al., 2014; Pradhan & Kumar, 2011; Samaei, Abbasion & Mirsayar, 2011; Sarrami-Foroushani & Azhari, 2014). So in this research, the values of small scale parameter μ are taken as zero up to 4 nm2 . Geometrical and material properties of the orthotropic SLGS are presented in Table 1. 6.1. Verification study To the best of authors’ knowledge no published literature is available for dynamic response of viscoelastic orthotropic SLGS resting on the orthotropic visco-Pasternak foundation. Since no reference to such a work is found to data in the literature, its verification is not possible. However, in an attempt to validate this study, a simplified analysis of this work is

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Table 1 Geometrical and material properties of the orthotropic SLGS (Analooei et al., 2013). E1 (GPa)

E2 (GPa)

G12 (GPa)

ν 12

ν 21

ρ (kg/m3 )

h (nm)

1765

1588

678.85

0.30

0.27

2300

0.34

0

-8

w*

-16

-24

Present,Metal Reddy(2000),Metal Present,Ceramic Reddy(2000),Ceramic

-32

-40

0

2.5

5

7.5 t*

10

12.5

15

Fig. 2. Comparison between deflection time-history of the centre of FG square plate under the uniform load.

done without considering the size effect, elastic foundation and orthotropic and viscoelastic properties of the nanoplate. The present results are compared with the work of Reddy (20 0 0) who analyzed functionally graded material (FGM) plate under a uniformly distributed load based on the FEM. For this purpose, the geometric properties are assumed to be: a = b = 0.2 m, h = 0.01 m and loading intensity is P0 = 106 N/m2 . Also, the central deflection and time are normalized as w∗ = wac2hPEm and

"

0

, where Em and ρ m are the corresponding properties of the metal. In Fig. 2, the results of the present method for a2 ρm the central deflection time-history have been compared with those obtained by Reddy (20 0 0) using finite element model for a simply supported FG square plate subjected to the uniformly load. As observed, in this case, our outcomes agree excellent with the finite elements results.

t∗ = t

Em

6.2. Parametric study For convenience, the following non-dimensional parameters are used in presenting the numerical results:

w∗ = CD =

w c hE2 , b2 P0 cd a2

#

ρ hD11

KW = ,

kw a4 , D11

g G= 2 a

$

D11 , ρh

KGξ = ∗

kgξ a2 , D11

t =t

$

KGη =

E2 , b2 ρ

kgη b2 D11

D11 =

E1 h3 12(1 − ν12 ν21 )

(28)

The values of the length to thickness ratio and aspect ratio are assumed to be 10 and 1, respectively, unless otherwise stated. Also, magnitude of the applied load is P0 = 106 N/m2 . Two sample problems are presented in this study. Firstly, the elastic nanoplate is assumed resting on the elastic medium and then the effects of nonlocal parameter, loading type, elastic foundation type and orthotropy angle of foundation on the transient response under dynamic load are investigated. Secondly, both of the structural and foundation are considered as viscoelastic material and the various influences on the dynamic response are studied. 6.2.1. The elastic nanoplate on the elastic foundation 6.2.1.1. Effect of nonlocal parameter and loading type. At first, the effect of nonlocal parameter on the dynamic response of orthotropic SLGS without foundation under sinusoidally and uniformly distributed transverse loads is depicted in Fig. 3(a) and (b), respectively. It is obvious that the nonlocal parameter μ is a significant parameter in the analysis of nanomaterials and should not be neglected in the nanostructure. It can be seen that the amplitude of deflection increases when the nonlocal parameter increases. It is need to point out that the zero value for nonlocal parameter (i.e., μ = 0) denotes the results obtained by the local elasticity theory which has the lowest deflection. This is because of the fact that nonlocal theory introduces a more flexible model wherein atoms are joined by elastic springs while the values of spring constants in local theory are assumed to be infinite. Therefore, the difference between predicted deflection by local and nonlocal

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a

b Fig. 3. Deflection history of an orthotropic SLGS for various nonlocal parameters under (a) sinusoidally and (b) uniformly loads.

theories is physically justifiable. Moreover, comparing Fig. 3(a) and (b) shows that the dynamic response for uniform loading is obtained more than sinusoidal loading. 6.2.1.2. Effect of elastic foundation. Fig. 4(a) and (b) demonstrates the effect of different elastic medium on the dynamic response of the orthotropic SLGS under sinusoidal load for μ = 1 and 3 nm2 , respectively. As can be seen, considering elastic medium decreases dynamic response of nanoplate. It is due to the fact that considering elastic medium leads to stiffer nanoplate. Furthermore, the effect of the Pasternak foundation is higher than the Winkler foundation for decreasing of the nanoplate. It is because Winkler foundation is capable to describe just normal load, while the Pasternak medium describes both normal and transverse shear loads. As shown, the Orthotropic Pasternak foundation is more effective than the isotropic Pasternak type to reduce the dynamic response of the nanoplate due to considering an arbitrarily oriented foundation. It is also concluded from Fig. 4(a) and (b) that with increasing the value of nonlocal parameter, the difference between the dynamic response of with and without foundation becomes more obvious. It is due to the fact that higher nonlocal parameter introduces a more flexible model. Consequently, existence of elastic foundation is an important factor for decreasing the dynamic response of the nanoplate and must be considered, especially in the higher nonlocal parameters. 6.2.1.3. Effect of orthotropy angle of foundation. The effect of orthotropy angle on the dynamic response of orthotropic nanoplate resting on orthotropic Pasternak foundation subjected to sinusoidal load is depicted in Fig. 5(a) and (b) for λ = 2 and 10, respectively, in which λ =

KGξ KGη

. However, it should be noted that for λ = 1 the foundation becomes isotropic Paster-

nak. The spring and η-shear constants are assumed as KW = 100 and KGη = 10, respectively, and the nonlocal parameter μ is considered to be 2 nm2 . Since the orthotropy angle of foundation can be affected the deflection of nanoplate, it is a significant factor. As may be seen from the figures, increasing orthotropy angle leads to decreasing of deflection. It can be

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107

a

b Fig. 4. Effect of the elastic foundation type on dynamic response of an orthotropic SLGS under sinusoidal load for (a) μ = 1 and (b) μ = 3 nm2 .

observed that as λ increases, the deflection amplitude decreases. Further, it is important to notice that the effect of the orthotropy angle is more evident for higher values of λ. Also, it can be concluded that θ = 45° is the best angle to obtain the minimum deflection among others. In the following examples, the effects of structural and foundation damping coefficient, length to thickness ratio, aspect ratio, loading type and modulus ratio on the transient response of viscoelastic orthotropic nanoplate resting on orthotropic visco-Pasternak foundation are investigated. For this purpose, it is considered that KW = 100, KGη = 10, KGξ = 20 and θ = 45°. Moreover, the non-dimensional structural damping coefficient G and the non-dimensional damping constant of foundation CD are taken to be 0.01 and 1, respectively (unless otherwise stated). 6.2.2. The viscoelastic nanoplate resting on the viscoelastic foundation 6.2.2.1. Effect of structural and foundation damping coefficient. To clarify the influences of damping coefficient of viscoelastic foundation and structural damping of the nanoplate on the dynamic response of the SLGS under a sinusoidal load, Fig. 6(a) and (b) is presented with μ = 2 and 4 nm2 , respectively. It can be observed that the dynamic response is significantly influenced by the structural damping coefficient G and the damping coefficient of the foundation CD . It is interesting to note that since increasing the structural and foundation damping causes more absorption of energy by the system, the dynamic response decreases. According to this significant difference, it is obvious that considering viscoelastic characterization both nanoplate and foundation can yield the accurate results with respect to non-visco ones. By comparing Fig. 6(a) and (b), it is found that increasing the nonlocal parameter increases amplitude and time interval of dynamic response. 6.2.2.2. Effect of length to thickness ratio. Fig. 7 displays the influence of length to thickness ratio a/h on the dynamic response of the embedded viscoelastic orthotropic SLGS subjected to sinusoidal load for μ = 1 nm2 . This figure demonstrates that decreasing the length to thickness ratio a/h reduces both the amplitude and the descending time obviously. In other

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a

b Fig. 5. Effect of orthotropy angle on the dynamic response of orthotropic SLGS resting on orthotropic Pasternak foundation under sinusoidal load for (a) λ = 2 and (b) λ = 10.

words, the nanoplate with lower a/h reaches the static state faster rather than those with higher a/h. Because as the length to thickness ratio decreases, the nanoplate becomes thicker thus the stiffness increases and consequently, the lateral deflection and time of dynamic response are decreased. 6.2.2.3. Effect of aspect ratio and loading type. The effect of aspect ratio a/b on the dynamic response of embedded viscoelastic orthotropic SLGS subjected to transverse sinusoidal and uniform loads is depicted in Fig. 8(a) and (b), respectively for μ = 1 nm2 . From both figures, it is observed that the deflection of the nanoplate increases with increasing of aspect ratio. As aspect ratio increases, the nanoplate is gradually converted to beam model and its rigidity reduces and therefore the deflection increases. Moreover, comparing Fig. 8(a) and (b) concludes that the amplitude of the SLGS subjected to uniform loading is more than sinusoidal loading. 6.2.2.4. Effect of modulus ratio. Finally, Table 2 illustrates the influence of modulus ratio Er on the dimensionless deflection of viscoelastic orthotropic SLGS resting on orthotropic visco-Pasternak medium subjected to sinusoidal for μ = 1 and 3 nm2 . The used material properties are taken as: Er = E1 /E2 varied, G12 = G13 = 0.6E2 , G23 = 0.5E2 , ν 12 = 0.25. It is seen that when the modulus ratio Er increases, the dynamic response decreases and consequently the viscoelastic orthotropic nanoplate reaches the static response much faster. For example, the orthotropic nanoplate with Er = 10 and Er = 30 for μ = 1 nm2 reaches the equilibrium state at t ∗ = 180 and t ∗ = 100, respectively. In addition, it can be found that the nonlocal parameter does not have obvious effect on the dynamic response orthotropic nanoplate with various modulus ratios. For instance, the orthotropic nanoplate with Er = 20 for both nonlocal parameters μ = 1 nm2 and μ = 3 nm2 approaches the static response at t ∗ = 140.

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a

b Fig. 6. Coupled effects of the damping coefficient of visco-Pasternak foundation and structural damping of the nanoplate under sinusoidal load on the dynamic response for (a) μ = 2 and (b) μ = 4 nm2 .

Fig. 7. Influence of length to thickness ratio on the dynamic response of the embedded viscoelastic orthotropic SLGS subjected to sinusoidal load for μ = 1 nm2 .

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a

b Fig. 8. Effect of aspect ratio on the dynamic response of embedded viscoelastic orthotropic SLGS under (a) sinusoidal and (b) uniform loads for μ = 1 nm2 . Table 2 Influence of modulus ratio on the dimensionless deflection of viscoelastic orthotropic SLGS resting on orthotropic visco-Pasternak medium subjected to sinusoidal. t∗

20 40 60 80 100 120 140 160 180 200

μ = 1 nm 2

μ = 3 nm2

Er = 10

Er = 20

Er = 30

Er = 10

Er = 20

Er = 30

0.2384 0.2362 0.2186 0.2211 0.2221 0.2218 0.2217 0.2218 0.2218 0.2218

0.1290 0.1128 0.1130 0.1132 0.1133 0.1134 0.1134 0.1134 0.1134 0.1134

0.086 0.0754 0.0765 0.0764 0.0764 0.0764 0.0764 0.0764 0.0764 0.0764

0.3064 0.2119 0.2363 0.2317 0.2318 0.2323 0.2320 0.2321 0.2321 0.2321

0.1125 0.1233 0.1182 0.1172 0.1174 0.1175 0.1175 0.1175 0.1175 0.1175

0.0800 0.0810 0.0786 0.0787 0.0787 0.0787 0.0787 0.0787 0.0787 0.0787

7. Conclusion The present research was concerned with the size dependent transient response viscoelastic orthotropic SLGS embedded in viscoelastic medium under uniform and sinusoidal dynamic loads, for the first time. The Kelvin–Voigt model was considered to describe the nanoplate viscoelasticity. The surrounding medium is simulated by orthotropic visco-Pasternak

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model. Based on the FSDT, the nonlocal motion equations were derived via Hamilton’s principle and then were solved analytically using Fourier series-Laplace transform technique for simply-supported boundary conditions. Considering the orthotropic properties of both nanoplate and foundation in conjunction with the structural damping and viscoelastic Pasternak foundation for the transient analysis is the main motivations of this paper. As it is presented, a detailed investigation of the effects of nonlocal parameter, length to thickness ratio, aspect ratio, elastic foundation type, structural damping, orthotropy directions and damping coefficient of the foundation, modulus ratio and loading type on the dynamic behavior of SLGS was carried out. The results indicated that presence of the elastic foundation increases the stiffness of the SLGS and therefore is a remarkable factor for decreasing the dynamic response particularly in the higher nonlocal parameters. It was also worth mentioning that orthotropic Pasternak medium with θ = 45° is more effective than others for decreasing the deflection. Moreover, it was found that increasing modulus ratio leads to decreasing the dynamic response of the orthotropic SLGS and consequently the orthotropic nanoplate reaches the equilibrium state much faster. In addition, it was found that increasing the values of structure damping and foundation damping coefficients causes an obvious decrease of amplitude and time interval of response. Furthermore, since the dynamic response was increased with increasing the nonlocal parameter, this effect is quite important and should not be neglected for analysis of nanostructures. The outcomes of this work were validated as far as possible by Reddy (20 0 0). Finally, it is hoped that the results presented here can be utilized as benchmarks for verifying results obtained from the other mathematical approaches and also would be beneficial for the design of NEMS devices. Acknowledgment The authors are grateful to University of Kashan for supporting this work by Grant No. 363443/64. They would also like to thank the Iranian Nanotechnology Development Committee for their financial support. Appendix A

%

&

E1 h ν12 E2 A¯ 2 = G12 h, A¯ 3 = h , + G12 , 1 − ν12 ν21 1 − ν12 ν21 % ν E & E2 h 12 2 + G12 , , B¯ 1 = h B¯ 2 = G12 h, B¯ 3 = 1 − ν12 ν21 1 − ν12 ν21 A¯ 1 =

C¯1 = Ks G13 h,

C¯2 = Ks G23 h,

D¯ 1 = −Ks G13 h, D¯ 6 = −I1 ,

E1 h3 , 12(1 − ν12 ν21 )

C¯4 = Ks G23 h,

D¯ 3 =

G12 h3 , 12

A¯ 5 = −I1

(A.1)

B¯ 4 = −I0 ,

B¯ 5 = −I1

(A.2)

C¯5 = −I0

(A.3)

D¯ 4 = −Ks G13 h,

D¯ 5 =

%

&

h3 ν12 E2 + G12 , 12 1 − ν12 ν21

D¯ 7 = −I2

E¯1 = −Ks G23 h, E¯6 = −I1 ,

D¯ 2 =

C¯3 = Ks G13 h,

A¯ 4 = −I0 ,

E¯2 =

(A.4) (A.4)

%

&

h3 ν12 E2 + G12 , 12 1 − ν12 ν21

E¯3 =

G12 h3 , 12

E¯4 =

E2 h3 , 12(1 − ν12 ν21 )

E¯5 = −Ks G23 h,

E¯7 = −I2

(A.5)

Appendix B The components of the matrices introduced in Eq. (25) can be expressed as follows: Components of the mass matrix:









M11 = A¯ 4 1 + μα 2 + μβ 2 , M12 = M13 = 0, M14 = A¯ 5 1 + μα 2 + μβ 2 , M15 = 0,     2 2 ¯ M21 = 0, M22 = B4 1 + μα + μβ , M23 = M24 = 0, M25 = B¯ 5 1 + μα 2 + μβ 2 ,   M31 = M32 = 0, M33 = C¯5 1 + μα 2 + μβ 2 , M34 = M35 = 0,     2 ¯ M41 = D6 1 + μα + μβ 2 , M42 = M43 = 0, M44 = D¯ 7 1 + μα 2 + μβ 2 , M45 = 0,     M51 = 0, M52 = E¯6 1 + μα 2 + μβ 2 , M53 = M54 = 0, M55 = E¯7 1 + μα 2 + μβ 2

(B.1)

Components of the damping matrix:





¯ C11 = −g A¯ 1 α 2 + A¯ 2 β 2 , C13 = C14 = C15 = 0, C12 =2 −αβ A23 g, ¯ ¯ ¯ C21 = −αβ B1 g, C22 = −g B2 α + B3 β ,    C23 = C24 = C25= 0, C31 = C32 = 0, C33 = −g C¯1 α 2 + C¯2 β 2 − cd 1 + μα 2 + μβ 2 , C34 = −αC¯3 g, C35 = −β C¯4 g,   2 2 ¯ ¯ ¯ ¯ ¯ C41 = C42 = 0, C43 = α D1 g, C44 = −g D2 α + D3 β − D4 , C45 = −αβ D5 g,   C51 = C52 = 0, C53 = β E¯1 g, C54 = −αβ E¯2 g, C55 = −g E¯3 α 2 + E¯4 β 2 − E¯5 ,

(B.2)

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Components of the stiffness matrix:





¯ K11 = − A¯ 1 α 2 + A¯ 2 β 2 ,  K12 2= −αβ A2 3 , K21 = −αβ B¯ 1 , K22 = − B¯ 2 α + B¯ 3 β , K31 = K32 = 0,





K33 = − C¯1 α 2 + C¯2 β 2 −

K34 = −αC¯3 , K35 = −β C¯4 , K41 = K42 = 0, K43 = α D¯ 1 , K51 = K52 = 0, K53 = β E¯1 ,





K13 = K14 = K15 = 0, K23 = K24 = K25 = 0, kgξ (cos2 θ α 2 + 2 cos θ sin θ αβ + sin θ β 2 ) 2 +kgη (sin θ α 2 − 2 sin θ cos θ αβ + cos2 θ β 2 ) − kw 2







1 + μα 2 + μβ 2 ,



¯ K44 = − D¯ 2 α 2 + D¯ 3 β 2 − D¯ 4 ,  2 K45 =2 −αβD5 , ¯ ¯ ¯ ¯ K54 = −αβ E2 , K55 = − E3 α + E4 β − E5 , (B.3)

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