Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials

International Journal of Engineering Science 109 (2016) 29–53

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Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials Mohammad Hosseini a,∗, Mohammad Shishesaz a, Khosro Naderan Tahan a, Amin Hadi b a b

Department of Mechanical Engineering, Shahid Chamran University, Ahvaz, Iran Mechanical Engineering Faculty, University of Tehran, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 29 July 2016 Revised 10 August 2016 Accepted 1 September 2016

Keywords: Nano-disk Strain gradient theory Stress analysis Functionally graded materials (FGMs) Size effect

a b s t r a c t This paper presents the stress analysis of rotating nano-disk made of functionally graded materials with nonlinearly varying thickness based on strain gradient theory. The equilibrium equation and corresponding boundary conditions of nano-disk were obtained using Hamilton’s principle. Because of the complexity of governing equations and boundary conditions, the equations are solved using numerical methods. Fixed boundary conditions are considered, in the numerical examples. This analysis is general and can be reduced to classical elasticity. The effect of various parameters such as graded index and thickness profile on stresses and high-order stresses were examined. Values of stresses at inner and outer radial are not zero, because stresses at inner and outer radius accumulate with stresses caused by strain gradient theory. Results show that the effects of thickness parameters are greater than the effect of graded index and the difference between the stress predicted by the classical theory and the strain gradient theory is large when the thickness of nano-disk is small. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Nanotechnology is the branch of technology that deals with dimensions and tolerances of less than 100 nanometers, especially the manipulation of individual atoms and molecules. Since the discovery of nanotechnology by Feynman (John, 1997), there are amounts of researches on properties of nano and micro structural elements by adopting experimental investigation, Molecular dynamics simulation and continuum mechanics approach. Both experimental and Molecular dynamics simulation results have shown that the small-scale effects have significant effect on mechanical properties of nano and micro structures (Daneshmehr, Rajabpoor, & Hadi, 2015). Experimental methods in nano scale are very expensive and difficult, molecular dynamic simulations are limited to structures with a small number of molecules and atoms. The continuum mechanics approach is less computationally expensive than molecular dynamic simulations. For solving this barrier, continuum theories are used. Classical continuum mechanics cannot predict small scale effect. Thus, several higher-order continuum theories were suggested to solve this problem. Among these theories, the couple stress theory (Toupin, 1962), Mindlin’s strain gradient theory (Mindlin & Eshel, 1968), nonlocal elasticity theory (Eringen, 1972, 1983, 2002), strain gradient theory (Aifantis, 1999), modified couple stress theory (Yang, Chong, Lam, & Tong, 2002) and modified strain gradient

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (M. Hosseini).

http://dx.doi.org/10.1016/j.ijengsci.2016.09.002 0020-7225/© 2016 Elsevier Ltd. All rights reserved.

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theory (Lam, Yang, Chong, Wang, & Tong, 2003) are the most popular and strong ones. Among the size dependent continuum theories, strain gradient theory initiated by Mindlin (Mindlin & Eshel, 1968) has been widely used to analyze many nanostructures problems. According to this theory in addition to the strain tensor, strain gradients are also considered in writing the strain energy density (Danesh & Asghari, 2014). Many researchers have applied the strain gradient theory for mechanical behavior of nano-sized structures such as beams, plate and shell (Akgöz & Civalek, 2016; Ansari, Gholami, Faghih Shojaei, Mohammadi, & Sahmani, 2015; Ansari, Gholami, & Norouzzadeh, 2016; Gholami, Darvizeh, Ansari, & Sadeghi, 2016; Hosseini & Bahaadini, 2016; Li & Hu, 2016; Li, Hu, & Li, 2016; Ojaghnezhad & Shodja, 2016; shahriari, Karamooz Ravari, & Zeighampour, 2015; Wang, Huang, Zhao, & Zhou, 2016; Zeighampour, 2015; Zeighampour & Tadi Beni, 2015; Zhou, Li, & Wang, 2016). Rotating disks in size of micron and sub-microns are of practical concern in many tools in micro/nanoelectromechanical systems (MEMS/NEMS), for example micro-gyroscopes (Tsai, Liou, Lin, & Li, 2009, Tsai, Liou, Lin, & Li, 2010, 2011) and micromotors (Lee, Kim, Bryant, & Ling, 2005). Mechanical behavior of micro-rotating disks based on the strain gradient elasticity is investigated by Danesh and Asghari (Danesh & Asghari, 2014). Functionally graded materials (FGMs) are a special group of heterogeneous composite materials with mechanical properties changing continuously from one surface to another (Nejad, Rastgoo, & Hadi, 2014b). A number of papers considering various aspects of FGM have been published in recent years (Anani & Rahimi, 2016; Calim, 2016; Golmakaniyoon & Akhlaghi, ´ 2016; Hadi, Rastgoo, Daneshmehr, & Ehsani, 2013; Heydarpour & Aghdam, 2016; Kiełczynski, Szalewski, Balcerzak, & Wieja, 2016; Nejad & Fatehi, 2015; Nejad, Rastgoo, & Hadi, 2014a, 2014b; S¸ ims¸ ek, 2016; Yang, Wang, & Lin, 2016; Zhang & Liew, 2016). Thanks to the advances in technology, FGMs have started to find their ways into micro/nanoelectromechanical systems (MEMS/NEMS) (Kahrobaiyan, Asghari, Rahaeifard, & Ahmadian, 2010; Nejad & Hadi, 2016a, 2016b; Nejad, Hadi, & Rastgoo, 2016; Zhang & Fu, 2012). Stress analysis of a functionally graded micro/nano rotating disk with variable thickness based on the strain gradient theory is studied by Baghani, Heydarzadeh and Roozbahani (Baghani, Heydarzadeh, & Roozbahani, 2016). The objective of this study is to obtain the elastic deformations and stresses of rotating nano-disk with nonlinear variable thickness made of functionally graded based on strain gradient theory. There are some problems with derivation of equations and results of above papers (Baghani et al., 2016). Given this problem, in this paper derivation of equations and results have been corrected. In addition, nonlinear function is assumed for profile and properties of disk.

2. Theory and formulation In this section, the strain gradient theory is formulated for nano-micro discs. In the classic elasticity theory, strain energy density function is dependent on the infinitesimal strain tensor, which is the symmetric part of gradient of displacement field u. In the strain gradient elasticity theory, second gradient of displacement field u appears in the equations of motion and boundary condition equations. The strain gradient elasticity theory introduces dilatation gradient tensor and the deviatoric stretch gradient tensor as well as the symmetric rotation gradient. Strain tensor ε and gradient of strain tensor ξ are:

 1 ∇ u + (∇ u )T 2  1  T = ∇ ε = ∇ ∇ u + (∇ u ) 2

ε= ξ

(1)

where

∇ = er

∂ 1 ∂ ∂ + eθ + ez ∂r r ∂θ ∂z

(2)

Therefore radially and the circumferential strains are:

⎡

∂u ⎢ ∂r ε=⎢ ⎣0 0

⎤ 0

0

u r 0

0⎦

⎥ ⎥

0

and components of second gradient of displacement field, u are:

(3)

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⎧ ∂ 2u ⎪ ⎪ξrrr = 2 ⎪ ∂r ⎪ ⎪   ⎪ ⎪ ∂ ur ⎪ u 1 ∂u ⎪ ⎨ξθ θ r = − 2 = r ∂r ∂ r  r u ∂ ⎪ u 1 ∂u r ⎪ ⎪ ξ = − = r θ θ ⎪ ⎪ r ∂r ∂ r  r2 ⎪ ⎪ u ⎪ ⎪ ⎩ξ = 1 ∂ u − u = ∂ r θ rθ 2 r ∂r ∂r r

(4)

The relationship between stress and strain is determined by using Hooke’s Law. Hook’s low is presented as follows for plain stress.

⎧ ⎨σr =

E (εr + υεθ ) 1 − υ2 ⎩σ = E (ε + υε ) r θ θ 1 − υ2

(5)

In general, the Hook’s low wrote as follows:

⎧ ⎪ ⎨σr = A ∂ u + B u ∂r r ⎪ ⎩σθ = B ∂ u + A u ∂r r

(6)

where

E 1 − υ2 Eυ B= 1 − υ2

A=

(7)

High-order stress tensor τ is defined as follows:

 ∂ u¯ 1  τi jk = τ jik = = a δ ξ + 2δ jk ξ ppi + δik ξ jpp + 2a2 δ jk ξipp ∂ ξi jk 2 1 i j kpp     + a3 δi j ξ ppk + δik ξ pp j + 2a4 ξi jk + a5 ξ jki + ξk ji .

(8)

where ai ’s and δ ij are the higher-order length scale material parameters and Kronecker’s delta, respectively. So by expanding the Eq. (8) and using Eq. (3), the high-order stress components are expressed as:

3

 ∂ 2u









1 3 ∂u 1 3 a + 2a2 + 2 a3 − a + 2 a2 + 2 a3 u 2 ∂ r r2 2 1 ∂ r2 r 2 1 1  ∂ 2u 1  1  ∂u 1  1  τθ θ r = a1 + a3 + a + a + 2 a + 2 a − a + a + 2 a + 2 a u 1 3 4 5 1 3 4 5 2 ∂ r r2 2 ∂ r2 r 2 1  ∂ 2u 1  1  ∂u 1  1  τr θ θ = a1 + a2 + a + 2 a + 2 a + 2 a − a + 2 a + 2 a + 2 a u 1 2 4 5 1 2 4 5 2 ∂ r r2 2 ∂ r2 r 2 1  ∂ 2u 1  1  ∂u 1  1  τθ r θ = a1 + a3 + a + a3 + 2 a4 + 2 a5 − a + a3 + 2a4 + 2a5 u 2 ∂ r r2 2 1 ∂ r2 r 2 1

τr r r =

a1 + 2a2 + 2 a3 + 2 a4 + 2 a5

+

(9)

The following simplifications are used in order to simplify the process of obtaining governing equations.

3 1 a1 + 2a2 + 2 a3 + 2 a4 + 2 a5 k4 = a1 + a3 + 2 a4 + 2 a5 2 2 3 1 k2 = a1 + 2a2 + 2 a3 k5 = a1 + a2 2 2 1 1 k3 = a1 + a3 k6 = a1 + 2 a2 + 2 a4 + 2 a5 2 2 k1 =

(10)

So Eq. (9) can be rewritten as follows:

⎡



k1

 ⎢ τr r r ⎢ τθ θ r = ⎢ ⎢k3 τr θ θ ⎣ k5

k2 r k4 r k6 r

⎤

k2 ⎧ 2 ⎫ ∂ u⎪ r 2 ⎥⎪ ⎪ ⎨ ∂ r2 ⎪ ⎬ ⎥ k4 ⎥ − 2 ⎥ ∂u , ⎪ r ⎦⎪ ⎪ ⎪ k6 ⎩ ∂ur ⎭ − 2 r −

τθ r θ = τθ θ r

(11)

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In order to obtain the governing equation and associated boundary conditions, Hamilton’s principle is used.



t2

t1

(δU − δ K − δW )dt = 0

(12)

where W, K and U are work done by the external loads, kinetic energy and total strain energy, respectively. Variation of total strain energy is defined as:



δU =

V

(σ : δε + τ : δξ )dV =

 2π  h ( r ) 

=

0

= 2π

0 ro

 ri

ro ri

 V



 σ jk δε jk + τi jk δξi jk dV

(σr δεr + σθ δεθ + τrrr δξrrr + τθ θ r δξθ θ r + τrθ θ δξrθ θ + τrθ r δξrθ r )rdrdzdθ

(σr δεr + σθ δεθ + τrrr δξrrr + τθ θ r δξθ θ r + τrθ θ δξrθ θ + τrθ r δξrθ r )rh(r )dr

(13)

and variation of work done by external energy is expressed as:

  δW = 2π rh(r ) σˆ r δ u + σˆ θ δv + τˆrrr δεrr + τˆrθ r δεrθ + τˆθ rr δεθ r + τˆθ θ r δεθ θ

(14)

Because v,ε rθ and ε θ r are zero, Eq. (14) is simplified as follows:

  δW = 2π rh(r ) σˆ r δ u + τˆrrr δεrr + τˆθ θ r δεθ θ      dδ u δu τˆθ θ r dδ u = 2π rh(r ) σˆ r δ u + τˆrrr + τˆθ θ r = 2π rh(r ) σˆ r + δ u + τˆrrr dr

r

r

(15)

dr

Kinetic energy of the rotating discs is defined as:

   1 1 ρ u˙ 2 dV → δ K = ρ (2u˙ δ u˙ )dV = ρ u˙ δ u˙ dV → 2 V 2 V V  2π  h ( r )  ro  ro δK = ρ u˙ δ u˙ dV = 2π ρ rh(r )u˙ δ u˙ dr K =

0

0

ri

ri

u˙ = (r + u )ωeθ → δ u˙ = (δ u )ωeθ →  ro δ K = 2π ρ rh(r )(r + u )ω (δ u )ωdr ri

u  r →r+uu→  ro δ K = 2π ρ r2 h(r )ω2 δ udr

(16)

ri

where ρ , ω and h(r) are density, angular velocity and thickness of rotating disk. ro and ri are the outer and inner radii of the micro-nano disk, respectively. By substituting Eqs. (13), (15) and (16) into (12) and using of Eqs. (3), (4) the governing equation and relative boundary conditions are obtained. Note that the displacement is only function of radius of disk.



⎛

ro

ri

⎞ ∂ 2δu dδ u + h r σ + τ + τ + τ ( ( ) ) r θθr rθ θ rθ r ⎝   ∂ r2  dr ⎠dr ( τθ θ r + τr θ θ + τr θ r ) 2 2 δu + h σθ − − ρr ω r    τˆ dδ u − 2π r h ( r ) σˆ r + θ θ r δ u + τˆrrr =0 r h τr r r

r

(17)

dr

Eq. (17) becomes simple by using integration by part and the variation principle.

 ri

ro



d ( h ( r σ r + τθ θ r + τr θ θ + τr θ r ) ) d2 (rhτrrr ) − +h dr dr2



+ r hτrrr − r hτˆrrr



 ∂δ u r | o + h ( r σ r + τθ θ r + τr θ θ ∂ r ri



σθ −

( τθ θ r + τr θ θ + τr θ r )



δ udr ω  τˆ d (rhτrrr ) + τr θ r ) − − rh σˆ r + θ θ r δ u|rroi = 0 r

dr

− ρr

2

2



r

(18)

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33

Thus, equilibrium equation and associated boundary conditions are achieved according to Eq. (18).

Equlibrium Equation : d ( h ( r σ r + τθ θ r + τr θ θ + τr θ r ) ) d2 (rhτrrr ) − +h dr dr 2 Boundary Conditions : rhτrrr − rhτˆrrr = 0 → τrrr = τˆrrr h ( r σ r + τθ θ r + τr θ θ + τr θ r ) −



σθ −

( τθ θ r + τr θ θ + τr θ r ) r



− ρ r2 ω2 = 0

@r = r1 , r2

  τˆ σˆ r + θ θ r = 0 @r = r1 , r2

d (rhτrrr ) − rh dr

(19)

r

Substituting Eqs. (6) and (9) into (19).

 3 d u + h ( 2k1 + k2 − k3 − k5 − k7 ) dr3  2  ⎛ ⎞ 2 d k1 dh dk1 d h d ( 2k1 + 2k2 − k3 − k5 − k7 ) ⎜−Ahr + r h dr2 + 2 dr dr + dr2 k1 + h ⎟ d2 u dr +⎝ ⎠ dr2 1 dh + ( 2k1 + 2k2 − k3 − k5 − k7 ) − h ( k2 + k3 + k4 + k5 + k6 + k7 + k8 )   2 r  ⎛ dr 2

d4 u r hk1 4 + dr

 

d k1 dh 2r h + k1 dr dr



⎞

dh dA d k2 dh dk2 d h + hr + Ah + h 2 + 2 + 2 k2 ⎟ du dr dr dr dr d r dr +⎝ ⎠ dr 1 dh 1 d ( 2k2 + k4 + k6 + k8 ) 1 − + 2 h ( 2k2 + k4 + k6 + k8 ) ( 2k2 + k4 + k6 + k8 ) − h dr  2r  r ⎛ r dr ⎞ dB 1 dh 1 d k2 dh dk2 d2 h 1 dh −B − h + Ah − h + 2 + k + 2 k + k + k + k ( ) 2 2 4 6 8 ⎟ ⎜ dr dr r r dr dr dr2 dr2 r 2 dr +⎝ ⎠u = ρ r2 ω2 h 1 1 d ( 2k2 + k4 + k6 + k8 ) + 2h − 3 h ( 2k2 + k4 + k6 + k8 ) dr r r

⎜− Ar

(20)

The boundary conditions are expressed as;

k1

1 d2 u 1 du + k2 − 2 k2 u = τˆrrr r dr dr2 r

(21)

 2 d u + h ( k1 + k2 − k3 − k5 − k7 ) dr 2    1 dk2 dh du + Ahr + h(k2 + k4 + k6 + k8 ) − h + k2 r dr dr dr      1 1 dk2 dh τˆθ θ r + Bh − 2 h(k2 + k4 + k6 + k8 ) + h + k2 u = rh σˆ r + @r = ri , ro

d3 u −rhk1 3 − dr



@r = ri , ro

dk1 dh r h + k1 dr dr



r

r

dr

dr

r

(22)

These equations must be dimensionless, in order to solve system of equations. Dimensionless parameters are defined as following:

k j (r )

r ro

k j (r ) =

u (r ) =

u (r ) ro

a j (r ) =

h (r ) =

h (r ) ro

A (r ) =

A (r ) Eo

B (r ) =

B (r ) Eo

r=

γ =

ρω2 ro2 Eo

Eoro2 a j (r ) Eoro2

j = 1, 2, ..., 8 j = 1, 2, . . . , 8

(23)

Therefore the dimensionless boundary conditions are expressed as:

1 d2 u¯ (r¯ ) 1 du¯ (r¯ ) τˆrrr k¯ 1 + k¯ 2 − 2 k¯ 2 u¯ = dr¯ ro Eo r¯ dr¯2 r¯

@r¯ = r¯i , 1

(24)

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M. Hosseini et al. / International Journal of Engineering Science 109 (2016) 29–53

and

−r¯h¯ k¯ 1





d3 u dr

−

3

+ Bh −

d k1 dh + k1 dr dr





+ h k1 + k2 − k3 − k5 − k7

 1  h k2 + k4 + k6 + k8 − r

+ Ahr +



r h

1 r

2



h k2 + k4 + k6 + k8



 h

d k2 dh + k2 dr dr





1 dk2 dh + h + k2 r dr dr

d4 u¯ + dr¯4

 

dk¯ 1 dh¯ ¯ k1 2r¯ h¯ + dr¯ dr¯







rh u= Eo

+ h¯ 2k¯ 1 + k¯ 2 − k¯ 3 − k¯ 5 − k¯ 7



2

  τˆθ θ r σˆ r + @r = r i , 1 ro r

dr¯3



⎞ d 2k¯ 1 + 2k¯ 2 − k¯ 3 − k¯ 5 − k¯ 7 d2 k¯ 1 dh¯ dk¯ 1 d2 h¯ ¯ ¯ ¯ ¯ ¯ ⎜−Ahr¯ + r¯ h dr2 + 2 dr¯ dr¯ + dr¯2 k1 + h ⎟ d2 u¯ dr +⎝ ⎠  1   dr¯2 dh¯  ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ + 2k1 + 2k2 − k3 − k5 − k7 − h k2 + k3 + k4 + k5 + k6 + k7 + k8 dr¯   2 r¯  ⎞ ⎛  2¯ ¯ ¯2 ¯ dk¯ 2 ¯ A h k h h d d d d d ¯ ¯ ¯ ¯ − r¯A¯ + r¯ h + A¯ h + h 2 + 2 + 2 k2 ⎜ ⎟ du¯ dr¯ dr¯ dr dr¯  dr¯ dr¯  +⎝ ⎠ ¯ ¯ ¯ ¯ ¯  1 d 2k2 + k4 + k6 + k8  dr¯ 1 dh  ¯ 1  − 2k2 + k¯ 4 + k¯ 6 + k¯ 8 − h¯ + 2 h¯ 2k¯ 2 + k¯ 4 + k¯ 6 + k¯ 8 dr ⎛ r¯ dr¯  r¯ ⎞ r¯ dB¯ ¯ 1 ¯ d2 k¯ 2 dh¯ 1 ¯¯ dh¯ dk¯ 2 d2 h¯ ¯ ¯ ⎜−B dr − dr¯ h + r¯ Ah − r¯ h dr2 + 2 dr¯ dr¯ + dr¯2 k2 ⎟ ⎜ ⎟ ⎜ ⎟ ¯ ρ ro2 ω2 ¯2 ¯ ¯  +⎜+ 1 dh 2k¯ + k¯ + k¯ + k¯ ⎟u = E r h = γ¯ r¯2 h¯ 2 4 6 8 ⎜ r¯2 dr¯  ⎟ o  ⎝ ⎠ 1 ¯ ¯ 1 ¯ d 2k¯ 2 + k¯ 4 + k¯ 6 + k¯ 8 + 2h − 3 h 2k2 + k¯ 4 + k¯ 6 + k¯ 8 dr r¯ r¯ ⎛

(25)

  d3 u¯





dr

du dr

and finally the Navier equation is expressed as:

r¯h¯ k¯ 1

  d2 u

(26)

In the case of constant thickness, these equations are simple as follows:

d4 u¯ r¯k¯ 1 4 + dr¯





 d3 u¯ dk¯ 1  ¯ 2r¯ + 2k1 + k¯ 2 − k¯ 3 − k¯ 5 − k¯ 7 dr¯ dr¯3 



⎞ d 2k¯ 1 + 2k¯ 2 − k¯ 3 − k¯ 5 − k¯ 7 d2 k¯ 1 2 −A¯ r¯ + r¯ 2 + ⎠ d u¯ dr dr +⎝  d r¯2 1 ¯ − k2 + k¯ 3 + k¯ 4 + k¯ 5 + k¯ 6 + k¯ 7 + k¯ 8 r¯  ⎞ ⎛  ¯ dA d2 k¯ 2 ¯ − r¯ +A + ⎜ ⎟ du¯ dr¯ dr2   +⎝ ⎠ dr¯ ¯ ¯ ¯ ¯   1 d 2k2 + k4 + k6 + k8 1 − + 2 2k¯ 2 + k¯ 4 + k¯ 6 + k¯ 8 dr r¯ ⎛ r¯ ⎞ dB¯ 1 ¯ 1 d2 k¯ 2 ⎜− + A − r¯ dr2 ⎟¯  +⎝ dr¯  r¯¯ u = γ¯ r¯2 ⎠ 1 ¯ 1 d 2k2 + k¯ 4 + k¯ 6 + k¯ 8 + 2 − 3 2k2 + k¯ 4 + k¯ 6 + k¯ 8 dr r¯ r¯ 2 ¯k1 d u¯ (r¯ ) + 1 k¯ 2 du¯ (r¯ ) − 1 k¯ 2 u¯ = τˆrrr @r¯ = r¯i , 1 dr¯ ro Eo r¯ dr¯2 r¯2   3 ¯   d2 u¯ d u¯ d k1 −r¯k¯ 1 3 − r¯ + k¯ 1 + k¯ 2 − k¯ 3 − k¯ 5 − k¯ 7 dr¯ dr¯ dr¯2 ⎛



 dk¯ 2 1 ¯ k2 + k¯ 4 + k¯ 6 + k¯ 8 − + A¯ r¯ + dr¯ r¯ 

+ B¯ −



 1 dk¯ 2 1 ¯ k2 + k¯ 4 + k¯ 6 + k¯ 8 + r¯ dr¯ r¯2

du¯ dr¯



r¯ u¯ = Eo

  τˆθ θ r σˆ r + @r¯ = r¯i , 1. ror¯

(27)

Because of the complexity of governing equations and boundary conditions, the equations are solved using numerical methods. In order to achieve proper answer, governing equation with two boundary conditions at the inner radius and two

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35

Fig. 1. Variation of the non-dimensional elastic modulus versus non-dimensional radius.

boundary conditions at the outer radius is resolved. In the next section, some of these results for different conditions are presented. 3. Results and discussion In this section, based on strain gradient theory, radial displacement, radial and circumferential stresses of nano-disk are presented for different parameters. In order to illustrate the size effect on behavior of functionally graded micro-nano disk, several numerical examples have been performed. It is proposed that the modulus of elasticity and density of the micro-nano disk material vary in the radial directions, as follows:

 r − r n

E ( r ) = ( Eo − Ei )

i

ro − ri

+ Ei ri < r < ro

(28)

 r − r n i ρ ( r ) = ( ρo − ρi ) + ρi r i < r < r o

(29)

ro − ri

where Ei , Eo , ρ i , and ρ o are modulus of elasticity at inner radii, modulus of elasticity at outer radii, density at inner radii and density at outer radii of disk, respectively. n called grading index. In this paper, the Poisson’s ratio is kept constant. Fig. 1 illustrates the variation of the non-dimensional modulus of elasticity through the non-dimensional radius. The value of n = 0 represents a fully metallic nano-disk, whereas infinite n indicates a fully ceramic nano-disk. The variation of the combination of ceramic and metal is linear for n = 1. Two models of the thickness profile are considered as follows.

⎧   r m  ⎪ ⎨h(r ) = hi 1 − q ri  r −m ⎪ ⎩h(r ) = hi ri

: model 1 (30) : model 2

Different forms of the thickness profiles for two models are shown in Figs. 2 and 3. If the size coefficients to be zero (a1 = a2 = a3 = a4 = a5 = 0), the classical elasticity is achieved. So equilibrium Eq. (26) and boundary conditions (24) and (25) are as follows:



 d2 u¯ A¯ h¯ r¯ + dr¯2





dh¯ dA¯ du¯ + r¯ h¯ + A¯ h¯ + r¯A¯ dr¯ dr¯ dr¯



B¯



ρ r2 ω2 2 ¯ 1 dh¯ dB¯ ¯ h − A¯ h¯ u¯ = − i + r¯ h = γ¯ r¯2 h¯ dr¯ dr¯ Ei r¯

(31)

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Fig. 2. Non-dimensional thickness profile of disk for model 1, h¯ i = 1, q = 0.415 and m = 3.

Fig. 3. Non-dimensional thickness profile of disk for model 2, h¯ i = 1.

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37

Table 1 Different value of variable used in the numerical study.

q m

a

b

c

0.96 0.5

0.80 1

0.4151965 3

Fig. 4. Comparison between classical theory and strain gradient theory for a functionally graded disk with variable thickness (n = 1.5).

and

du¯ 1 u¯ A¯ + B¯ = dr¯ Ei r¯

 σˆ r +

τˆθ θ r ri r¯

 @r¯ = r¯i , 1

(32)

The following properties considered for a functionally graded rotating disk with variable thickness and comparison with other works (Bayat, Saleem, Sahari, Hamouda, & Mahdi, 2008).

Ei = EAl = 70GPa Eo = ECer = 151GPa kg ρi = ρAl = 2700 3 ρo = ρCer = 5700 kg m m3 ro = 1 × 10−9 m ri = 0.2 × 10−9 m

υ = 0.3

(33)

In the first example it is assumed that elastic modulus, density and thickness of micro-nano disk vary according to Eqs. (28), (29) and (30), respectively. The following values considered for the variables, to compare this study with other studies (Table 1). In order to verify the accuracy and reliability of the present work, when a1 = a2 = a3 = a4 = a5 for model 1 are neglected, a comparison of the result of this paper with Bayat et al., 2008, as shown in Figs 4-6. It can be seen that there is an excellent agreement between the results obtained in this paper and those reported in Bayat et al., 2008.

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M. Hosseini et al. / International Journal of Engineering Science 109 (2016) 29–53

Fig. 5. Comparison between classical theory and strain gradient theory for a functionally graded disk with variable thickness (q = 0.96, m = 0.5).

Fig. 6. Comparison between non-dimensional circumferential stress of classical theory and strain gradient theory for a functionally graded disk with variable thickness (n = 1.5).

M. Hosseini et al. / International Journal of Engineering Science 109 (2016) 29–53

Fig. 7. Dimensionless radial stress changes for different values of q (m = 0.5, n = 1.5).

Fig. 8. Dimensionless circumferential stress changes for different values of q (m = 0.5, n = 1.5).

39

40

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Fig. 9. The effect of q on the dimensionless τ rrr (m = 0.5, n = 1.5).

Fig. 10. The effect of q on the dimensionless τ θ θ r (m = 0.5, n = 1.5).

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Fig. 11. The effect of q on the dimensionless τ rθ θ (m = 0.5, n = 1.5).

Fig. 12. The effect of n on the dimensionless radial stress (q = 0.96, m = 0.5).

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Fig. 13. The effect of n on the dimensionless circumferential stress (q = 0.96, m = 0.5).

Fig. 14. The effect of n on the dimensionless τ rrr (q = 0.96, m = 0.5).

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Fig. 15. The effect of n on the dimensionless τ θ θ r (q = 0.96, m = 0.5).

Fig. 16. The effect of n on the dimensionless τ rθ θ (q = 0.96, m = 0.5).

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Fig. 17. The effect of m on the dimensionless radial stress (q = 0.96, n = 1.5).

Fig. 18. The effect of m on the dimensionless circumferential stress (q = 0.96, n = 1.5).

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Fig. 19. The effect of m on the dimensionless τ rrr (q = 0.96, n = 1.5).

Fig. 20. The effect of m on the dimensionless τ θ θ r (q = 0.96, n = 1.5).

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Fig. 21. The effect of m on the dimensionless τ rθ θ (q = 0.96, n = 1.5).

In the case of nano-disk a¯ i = 0. So, the coefficients a¯ i s are considered as follows:

⎧ a¯ 1 (r ) = 0.003 ⎪ ⎪ ⎨a¯ 2 (r ) = −0.008 a¯ 3 (r ) = 0.144 ⎪ ⎪ ⎩a¯ 4 (r ) = 0.098 a¯ 5 (r ) = 0.263

(1)

Some of the numerical results have been obtained and presented in Figs. 7–31 to explain the behavior of functionally graded rotating nano-disk with variable thickness. Figs. 7 and 8 illustrate the dimensionless radial and circumferential stresses when the thickness of the disk is changed according to model 1. These two figures show that by increasing the amount of q up to 0.25 stresses increase, but then stresses reduced and the least amount of stresses occurs for q = 1. Also, absolute value of stresses increase as dimensionless radial increases for all of q. Values of stresses at the inner and outer radial are not zero, because classical stresses at inner and outer radius accumulate with stresses caused by strain gradient theory. Figs. 9-11 Show high-order stresses of rotating functionally graded nano-disk for model 1. By increasing the amount of q up to 0.25 absolute of high-order stress τ rrr decreases, but then high-order stress τ rrr increases. This fig. represents one of boundary conditions that is satisfied at inner and outer radii. The minimum value of τ rrr occurred for q = 0.25, while the minimum value of high-order stresses τ θ θ r and τ rθ θ occurred for q = 1. In all cases, absolute value of high-order stresses τ θ θ r and τ rθ θ are reduced with increasing radius. Change of dimensionless stresses versus variation of graded index (n) show in Figs 12 and 13. It can be seen that absolute value of radial and circumferential stresses decrease as n increases, but change of n has a significant influence on nondimensional stresses. The changes of the high-order stresses for model 1 versus to r¯ for various values n are shown in Figs. 14-16. From these figures, we know that the absolute value of τ rrr , τ θ θ r and τ rθ θ decrease with the increasing of the gradient indexes. The dimensionless high-order stresses are more sensitive to q than n. So the effect of thickness profiles is greater than the effect of graded index. The maximum value of dimensionless stress τ rrr occurred near the outer radius, while the maximum value of dimensionless stress τ θ θ r and τ rθ θ occurred at inner radii. The effect of m on the dimensionless radial, circumferential stresses and high-order stresses for model 1 has been shown in Figs. 17-21. It is clearly observed from these figures that the absolute values of radial and circumferential stresses increase as m increases. Values of radial stress for each value of m, n and q at the inner and outer radial are not zero, because stresses

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Fig. 22. The effect of n on the dimensionless radial stress (m = 0.5).

Fig. 23. The effect of n on the dimensionless circumferential stress (m = 0.5).

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Fig. 24. The effect of n on the dimensionless τ rrr (m = 0.5).

Fig. 25. The effect of n on the dimensionless τ θ θ r (m = 0.5).

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Fig. 26. The effect of n on the dimensionless τ rθ θ (m = 0.5).

Fig. 27. The effect of m on the dimensionless radial stress (n = 1.5).

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Fig. 28. The effect of m on the dimensionless circumferential stress (n = 1.5).

Fig. 29. The effect of m on the dimensionless τ rrr (n = 1.5).

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Fig. 30. The effect of m on the dimensionless τ θ θ r (n = 1.5).

Fig. 31. The effect of m on the dimensionless τ rθ θ (n = 1.5).

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at inner and outer radius accumulate with stresses caused by strain gradient theory. But non-dimensional stress τ rrr is zero for each value of m, n and q at inner and outer radius, because τ rrr is one of boundary conditions and must be satisfied. From Fig. 19, it is prominent seen that the influences of the two parameters m and dimensionless radius on nondimensional stress τ rrr . In this figure, dimensionless stress τ rrr increases as m increases and m has a significant influence on τ rrr . Also, in this case, the maximum value of stress τ rrr occurs near the outer radius for all samples. The dimensionless high-order stresses τ θ θ r and τ rθ θ predicted by the strain gradient theory shown in Figs. 20 and 21. It is apparent that τ θ θ r and τ rθ θ non-monotonically vary with m or dimensionless radii. Figs 22 and 23 illustrate the dimensionless radial and circumferential stresses when the thickness of the disk is changed according to model 2. The maximum radial and circumferential stresses occur at inner radius. The changes of the high-order stresses for model 2 versus to r¯ for various values n are shown in Figs. 24-26. The maximum higher-order stresses τ rrr occur near the inner radius for all samples. It is seen that change of n does not have a significant effect on non-dimensional stresses. The maximum value of higher-order stresses τ θ θ r and τ rθ θ occur at the inner radius for all samples and high-order stresses decrease as dimensionless radius increases. Figs. 25 and 26 depict that change of n does not have a significant influence on non-dimensional stresses. The changes of the stresses and high-order stresses for model 2 versus to r¯ for various values m are shown in Figs. 27-31. From these figs, we can observe that the m plays important role in the stress analysis of FG rotating nano-disk. Moreover, non-dimensional stresses decrease as m increases when m > 0.1. Again for these cases the effect of thickness profiles is greater than the effect of graded index. 4. Conclusion The strain gradient theory predicts a different stresses than the classical elasticity theory, because there are four boundary conditions for strain gradient theory, while there are two boundary conditions in classical elasticity theory. Values of stresses at inner and outer radii are not zero, because stresses at inner and outer radius accumulate with stresses caused by strain gradient theory. When the thickness varies according to model 1, maximum value of high-order stress τ rrr occurred near the outer radius. While the thickness varies according to model 2, maximum value of high-order stress τ rrr occurred near the inner radius. high-order stresses τ θ θ r and τ rθ θ occurred at inner radius of nano-disk for all thickness profiles. The dimensionless stresses and dimensionless high-order stresses are more sensitive to q than n. 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