A nonlocal strain gradient theory for porous functionally graded curved nanobeams under different boundary conditions Текст научной статьи по специальности «Физика»

УДК 539.3

Нелокальная теория градиента деформации для изогнутых нанобалок из пористого функционально градиентного материала при различных граничных условиях

A.M. Zenkour1,2, A.F. Radwan3

1 Университет короля Абдулазиза, Джидда, 21589, Саудовская Аравия 2 Университет Кафр-эш-Шейха, Кафр-эш-Шейх, 33516, Египет 3 Нильский научно-технический институт, Кафр-эш-Шейх, 33514, Египет

С использованием нелокальной теории градиента деформации изучено поведение изогнутых нанобалок из пористого функционально градиентного материала при изгибе, продольном изгибе и свободных колебаниях с различными граничными условиями. В указанной теории напряжения зависят от градиентов деформации, содержатся нелокальные параметры и параметры линейных размеров материала. Для описания поля смещений использована теория синусоидальной сдвиговой деформации балки с тремя переменными, которая не требует введения коэффициента корректировки сдвига. С помощью нелокальной теории градиента деформации проведен учет влияния жесткости упрочнения и разупрочнения для исследуемых нанобалок. Изменение свойств пористого функционально градиентного материала изогнутых нанобалок по толщине происходит в соответствии со степенной моделью. На основе принципа Гамильтона для них получены определяющие уравнения. Проведено сравнение полученных численных результатов с имеющимися данными для идеальных функционально градиентных изогнутых нанобалок. Исследовано влияние параметра градиента деформации, угла раскрытия, нелокального параметра, граничных условий, показателя степенного закона, а также коэффициента пористости на изгиб, продольный изгиб и частоту свободных колебаний идеальных и пористых функционально градиентных изогнутых нано-балок.

Ключевые слова: пористость, функционально градиентные изогнутые нанобалки, изгиб, продольный изгиб, свободные колебания, граничные условия, нелокальная теория градиента деформации

DOI 10.24411/1683-805X-2020-13008

A nonlocal strain gradient theory for porous functionally graded curved nanobeams under different boundary conditions

A.M. Zenkour1,2 and A.F. Radwan3

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt 3 Department of Mathematics and Statistics, Higher Institute of Management and Information Technology, Nile for Science and Technology, Kafrelsheikh, 33514, Egypt

This paper investigates the bending, buckling and free vibration behaviors of porous functionally graded curved na-nobeams with different boundary conditions via a nonlocal strain gradient theory. The stresses are dependent on the strain gradients according to the nonlocal strain gradient theory. This theory contains both nonlocal and material length-scale parameters. The three-variable sinusoidal shear deformation beam theory is used to describe the displacement field and do not need any shear correction factor. The nonlocal strain gradient theory is employed to capture both hardening and softening stiffness influences on the present nanobeams. The material properties for the present porous functionally graded curved nanobeams are varying through-the-thickness due to the power law model. Hamilton's principle is applied to obtain the governing equations of porous functionally graded curved nanobeams. Numerical results are validated by comparison with the corresponding ones of perfect functionally graded curved nanobeams in the literature. The effects of the strain gradient parameter, opening angle, nonlocal parameter, boundary conditions, power-law index, porosity factor on the bending, buckling and free vibration frequencies of perfect and porous functionally graded curved nanobeams are all investigated.

Keywords: porosity, functionally graded curved nanobeams, bending, buckling, free vibration, boundary conditions, nonlocal strain gradient theory

© Zenkour A.M., Radwan A.F., 2020

1. Introduction

Nanostructures are widely used in industry and science. They lead to lose their reliability because of their pronounced interface which appears with unexpected variation in material properties [1]. To deal with this issue, we need to introduce a functionally graded material (FGM) that has a silky variation in its material properties. Functionally graded material has characteristics that have made them be largely used in different shapes in multiple industries [2-4]. Also, the nanostructures are greatest small and have fully one-dimension in the framework of nanometers and these are formed of the nanosized structural elements. For these specific characteristics of particles on the nano-beams, curved nanobeam, nanoscales, nanorods, na-noplates, nanoshells, and nanorings have a lot of applications in nanotechnology and microscopic devices [5-7]. Radwan and Sobhy [8] used the nonlocal strain gradient theory to discuss the dynamic variation of a single-layered graphene sheet on a visco-Pasternak foundation under a time-harmonic thermal load and different boundary conditions. Compressive studies for beams and plates are discussed by many authors [9-17].

The buckling loads and vibration frequencies of nanobeams are of enormous considerable interest work and have more applications in micro- and nano-scale devices [18-23]. Related applications of nonlocal strain gradient theory are concealment buckling of size-dependent beams under simply supported boundary condition [24], where the governing equations are deduced by Hamiltons principle and also, the boundary conditions are given. In the framework of nonlocal strain gradient theory, Li et al. [25] investigated the free vibration of functionally graded nanobeams with the size-dependent Timoshenko's beam model. Based on both nonlocal strain gradient theory and Euler-Bernoulli's beam theory, Simsek [26] discussed a nonlinear vibration of a functionally graded nanobeam with immovable ends using a novel size-dependent model. Allam and Radwan [27] investigated a comprehensive study for viscoelastic functionally graded curved nanobeam embedded in an elastic substance using the nonlocal strain gradient theory.

The small-scale effects cannot be described by the traditional classical elasticity. Some theories have been developed to capture such effects in continuum mechanics, like the modified couple stress model [28], the nonlocal Eringen's theory [29], the strain gradient theory [30], the micropolar theory of Eringen [31] and others. The scaling effect of long-range force with totally no investigation of the strain gradient is con-

sidered by nonlocal Eringen's model. The forces between internal length-scale and atoms in this theory are considered in construction of constitutive relations. Also, the stress at a point counts on the strain at all points in the body according to nonlocal Eringen's model [32-34]. The equilibrium differential equations for the bending response of functionally graded curved beam and nanobeam are discussed and solved numerically in Refs. [35, 36]. However, the buckling load of a functionally graded curved beam is achieved in [37, 38] and the vibration of a functionally graded curved beam is achieved in Refs. [39, 40].

Porosity (void fraction) is a fraction of the volume of voids through the total volume, between 0 and 1 and it is a measure of void spaces in the material [4144]. For functionally graded structures with porosities, many investigators have discussed the vibration behavior [45-49], the buckling response [50-53] and both buckling and vibration [54-56]. Radwan [57] investigated a thermomechanical buckling and vibration of porous functionally graded nanoplates resting on elastic foundations using 2D and quasi-3D integral theories.

As given in the previous studies and to the best of authors' knowledges, up to now, the porosity influences on the bending, buckling and vibration of the functionally graded curved nanobeams have not been communicated in the literature.

The bending, buckling and vibrations of porous functionally graded curved nanobeams using the nonlocal strain gradient theory under different boundary conditions are studied. The material properties for the present curved nanobeams are graded in the radial direction.

The dynamic equations of the porous functionally graded curved nanobeams using the nonlocal strain gradient theory are derived by Hamilton's principle. The outcomes of buckling are compared with that predicted for a perfect functionally graded curved nano-beam in the literature.

Several examples with numerical data are introduced to explain the influences of the porosity factor, power-law index, nonlocal and material length scale parameters on the behavior of functionally graded curved nanobeams.

2. Theories and relations

2.1. Porous functionally graded curved nanobeam

Consider a porous functionally graded curved nanobeam with thickness h and length L as displayed in Fig. 1. The effective properties of the porous functionally graded curved nanobeam vary continuously

-9_ 1 f_W6 - Ws + — 09

( дУ д2Wb \ "д9 + Ш2

R

_i _ 1

=9_ R2

(4)

_2_ д2Ws о _-1W

I2 ' l9z _ R д9

R2 д92 Ф(z) _ 1 -

( z)

dz '

Fig. 1. Schematic of porous functionally graded curved nanobeam

Hamilton's principle is given to derive the dynamic equations for porous functionally graded curved nanobeam as follows:

J5(n s +nw

П * )dt _ 0,

(5)

through the thickness. Let the material properties with

° . , „ „ JT . ,, where n, nW and n are strain energy, work done by

a p°rosity volume function, _a of porous toctwraUy external forces and kinetic energy respectively, and

graded curved nanobeam, like Young's modulus E. shear modulus G = E/[2(1 + v)] and the material density p, are determined as follows:

'2 z + h^y

P ( Z ) _ Pm

Pc

1 -

2 z

2h

i

a "2"

2h

(1)

in which P = E, p and the metal property is Pm, also the ceramic property is Pc, and y is the power-law index. For the perfect functionally graded curved nanobeam a = 0.

are defined as

0ПS _ J(a99Se99 + CT9zSe9z)dv,

P

о

B0 д(Wb + Ws )

R

д9

x^iWlW. -b + w■ )

д9

d9,

(6)

snw _Jp( z)[i)9dv9+(wb+ws )

V

xS(Wb +Ws )]dv, h _ R + h,

2.2. Kinematic relations

in which B0 is the buckling load. By incorporating

Eqs. (6) into Eq. (5), then the equations of motion can The three-unknown sinusoidal shear deformation be deduced as

beam model is used here to depict the displacement components (Ue, Wr) as

(2)

U9 (9, r ) + R у (9, t ) z дWb (9, t) ¥(z) дWs (9, t)

R

д9

R

д9

Wr(e, r) = -w 6 (e, t) - ws (e, t),

where Ue and Wr are the tangential and radial displacements,

¥(z) = z-—sin| —z % V%

According to the displacement model (2), the nonzero strain components are given as

eee = 4 + zeXe +¥ (z) e2, eez = a(z)y|L, (3)

where

1 дЖ9 + ^ дМ9 _ д2У т

' ' - T1 " + T 2

R д9 R2 д9

дt2

д 3Wb дt 2д9

+I

д 3Ws

дt 2д9

- - n9 R 9

1 д2M9 B0 д2(Wb + Ws)

R2 д92

R2

- qh _ I

д3У 2 дt 2д9

+1

д92 д 4Wb 4 дt2д92

д4Ws - д2(Wb + ws) 5 -.2^2 T 0 Я(2 :

—N9 R 9

дгд92 дt2

1 д2^9 B0 д2(Wb + Ws)

R2 д92

R2

д9

2

1 Q - qh _ т _д3У.+i.ê4W1

R д9 У 3 2д9 5 2д92

т д4Ws - д2(Wb + Ws) ^16 ,2^2 10

2д92

дt2

(7)

in which

h/ 2

= J P(z)

-h/ 2

[Л, 12,13]

1+£ Z ^if) R, R, R

M ц aMe _f-, 5 52 \B 80 + A 1 + A _2] Me „9 _ 1 ^ „9 [B11se + Ai2se+ A22se],

r2 ш2

R2 Ш2

1+R1 dz •

^e-

Ц

R2 ae2

2

1 -

5 a

r2 ae2

(14)

h 2

(8)

R

dz.

[/4, /5] = J p(z)[z, ¥(z)]-^dz

-hi 2 R

h 2

[/0, /6] = J p(z)

-h 2

The stress resultants and forces given in Eq. (7) can be written as

h 2

{Ne, M0, Se} = J {1, z, ¥(z)}aeedz, -h/ 2 h 2

Q.e = J ^(z)aezdz.

-V 2

2.3. Nonlocal strain gradient theory for porous functionally graded curved nanobeams

x [ ^ + A22se + A33s2],

0e-5 a2 Л

r2 ae2

r 2 ae2

a44ï ez

in which, the coefficients B- and A- can be written as

[A11, B11, A12, B22, A22, A33]

h 2

_ J E(z)[1, z, z2, ^(z), z^(z), ^2(z)]dz, (15)

(9)

-h 2

h/ 2

A44 _ J G(z)Ф2(z)dz.

-hf 2

Applying operator

2

1 -

ц a

r 2 ae2

According to the nonlocal strain gradient theory „ , „

that given by Aifantis [58] and Lim et al. [59], the to Eqsf(7) and using Eqsu(4) and (14) then, the equa-

stress tensor a- is introduced as _ c(°).

а- _cw-V-cf

ij - -

.(1)

(10)

tions of motion are given by

f a 5 a3 л

ae r2 ae3

where Cy(0) and c!y(1) are stress tensor of nonlocal and high-order nonlocal, respectively. By the nonlocal , A12

AA1+B1 if-w ь - w -

R2 R3

Eringen's theory [28, 29], the nonlocal stresses а and Gy(1) are given by

(0)

R3

in which

(1 - mV2)а(0) _ClJpqspq, (1 -mV2)аf _42CmVspq,

1 a2

(11)

1 -

R

ц a

r 2 ae2

au a2w

b \

V

2 Л

ae ae

" a2u

B22 + A22 R3 R4

ae

a 22

ae2

1

at2

+12

1-

r 2 ae2

2

R

V2 _

(

r 2 ae2

and spq, Cijpq and Vspq are nonlocal parameter, strain gradient length scale parameter, strain tensor, elastic coefficients and first order strain gradient, respectively. Applying operator (1 - ^.V2) to Eq. (10) and by using Eq. (11), one can get

(1 -^V2)Cj = (1 -^V2) C,JpqSpq, (12)

where = Equation (12) can be written as follows:

B

B11 _a_

r2 ae2

ч 2 Л

a 3w b aeat2

2 Л

+1

aw

aeat2

-wb - ws +

au ae

11 R

a12 az if au . a2wb Л

r2 ae2

ae ae2

f b22 a22 a2 ia2w-

R

r2 ae2

ae2

r

2

1-

ц a

r 2 ae2

b0 a2(Wb+w-) T a3u

R2

ae2

ц a2 Л I Ce

v1 r 2 ae2y

'ez

1 -

л

r 2 ae2

E ( z)eee G ( z )<Eez

.(13)

a 4w-ae2at2

-10

aeat

a 2(Wb + w- )

2 +14

a 4wb ae2at2

(16)

Substitute Eq. (3) into Eq. (13) and using expressions Eq. (9) one obtains:

i , \

2 Л

1 -

5 a

r 2 ae2

7

Д11 R

at2

BZL aLЛ

r2 ae2

-qh

_ 0,

-wb - w- +-aU ae

r2 ae2

r2 ae2

[ A11s e + B11se + B22s e],

B,

11 R

a22 a2 if au a2wb Л

22_

r2 ae2

ae ae2

B22 , A33 д

2 Ля2

R R2 д92

д2Ws A44 д2W'

д92

R2 д92

ц д

R2 д92

2

B0 д2(Wb + Ws ) . д3U

R2

2

v

7 д4Wb т д4Ws 7 д2(Wb + ws) г I5—+16—-10—5-- + qh

д92

+13

д9дt2

5 дв2дt2

д92дt2

дt2

_ 0.

3. Solution procedure

Analytical solution for bending, buckling and vibration of porous functionally graded curved nanobeams under different boundary conditions are performed. In this section, the dynamic form of the governing equations (16) are solved analytically by the Navier method. In this method, the generalized displacements У (9, t ), Wb (9, t ) and Ws (9, t ) are given as

У (9, t ) _ Un e^, n d9

Wb (9, t) _ WbT(9)emn, Ws (9, t ) _ W^T (9)ei(ant,

(17)

where (Un, Wn , Wni) are unknown functions and t (e) is a function which is selected as the porous functionally graded curved nanobeam eigenfunction and given as [60]:

S-S: t (e) = sin ^ n%e

C-S: t(9) _ sin I —9

n%,

cos| I-1

C-C: t(9) _ sin21 —9

(18)

F-F: t(9) _ cos21 y9

sin2In^9I +1

P

Three different boundary conditions are introduced in this problem as Wb = Ws = M = S0 = 0 at 0 = 0, p for the simply-supported boundary condition, U = Wb = Ws = 0 at 0 = 0, p for clamped boundary condition and M0 = Q0 = 0 at 0 = 0, p for free boundary condition. Substitute Eq. (17) into Eq. (16) one obtains:

P11 P12 P13 ■ Un ' 0'

P12 P22 P23 < wb _ q9 , 5 (19)

P13 P21 P33 _ .q9.

where the coefficients Pj for porous functionally graded curved nanobeam are given as follows:

P11 _ Л®2^ +ТТ(-I1R W + A11R2 + 2 BnR R

+ A12)L3 -4(A11R2 + 2B11R + Al2)£5, R

P12 —-¿-(-i 2 R3®2 + A11R+B11L

R3

+ -U-12 R^®2 + AnR^ + BnR 2 R

+ A12 R + Bn^)L3BnR + A12)£5, R

P13 = -^{~hR*®n + AnR + Bn)L

R3

T(-13 R3M®n + AnR^ + B22 R 2

A22R + Bn^)L3-^-(B^R + A22)£5, R

P21 (-12 R3®2 + A11R+Bn)L2

R3

+ -^(-12 R3m®2 + AUR^ + B11R 2 R

+ A12R + Bn^)L4 B11R + A12K6, R

P22 _ (-I0®2+A11R "X +é-(i4R 4®n

R

+10R Wn - R2B0 - An^ - 2BnR)L2

1 3 2 (20) + ^(-14R M®n + iR0 + iA12R

R

+ 2iB11^)L4 + ^iA12L6,

R

P23 _4(-i0 R 2®n+An)£0 - ~~r(-15 R 4®2n R R

-10 R 2ц®П + R2 B0 + An^ + BnR + B22 R)L

+ -^(-I5 R3^®n + RM^0 + A22 R + Bn£, R

+ B22^)L4 A22L6,

R

P31 _4-( i3R3®n - AnR+Bu)L2

R3

+ 13R3^®n + AnR^ + B22 R 2 R

+ A22R + Bn^)L4 -Jg-^R + A22)L6, R

Table 1. Effects of porosity factor, various opening angles, length scale and nonlocal parameters on the buckling load of S-S perfect and porous functionally graded curved nanobeams (L/h = 10, y = 1)

4 Theory H = 0 H 1 H 2

P = rc/3 P = rc/2 P = rc/3 P = rc/2 P = rc/3 P = rc/2

Ref. [37] 5.1234 1.6111 4.6631 1.4664 4.2788 1.3455

0 Present a = 0.0 5.1205 1.6102 4.6605 1.4656 4.2763 1.3453

a = 0.1 4.5706 1.4365 4.1601 1.3075 3.8172 1.1997

Ref. [37] 5.6290 1.7702 5.1234 1.6111 4.7011 1.4783

1 Present a = 0.0 5.6258 1.7692 5.1205 1.6102 4.6984 1.4775

a = 0.1 5.0217 1.5783 4.5706 1.4365 4.1939 1.3181

Ref. [37] 6.1347 1.9292 5.5836 1.7559 5.1234 1.6111

2 Present a = 0.0 6.1312 1.9281 5.5804 1.7549 5.1205 1.6102

a = 0.1 5.4729 1.7201 4.9812 1.5656 4.5706 1.4365

Ref. [37] 6.6403 2.0882 6.0438 1.9006 5.5457 1.7439

3 Present a = 0.0 6.6366 2.0870 6.0404 1.8995 5.5425 1.7429

a = 0.1 5.9239 1.8619 5.3918 1.6946 4.9474 1.5549

P32 10RX + A11)L0 I5R4«2

R R

- I0R V« + R2 B0 + An4 + B11R + B22 R)L

+ 15 R V«2 + MRB0 + A22 R R

+ B114 + B224)L4 - -T6 A22L6,

R

P33 = -U10 R 2«2 + AOL -4(-i6 R 4«2

R2

R4

-10 R V« + A44 R 2 + R2 B0 + An4 + 2 B22 R)L

?(-i6 r3h«2 + A44 R4 + ^RB0 + A33R

+2B22Vl4 --=6 A33L6' R

q0h(urc2n2 + R2P2) . f

—sin I —0

R2P2

P

where

L = J r (e)r (6)de5 £1 = J ^^de,

L2=P d!Tie) t (e)de, L3=P dTe. dreeide,

de2

de3 de

P

£4 =J ^ r (e)de, £5 =J

P d5

de

d5r (e) dT (e) de5 de

L6 = } ^ r (e)de.

de6

Therefore, the bending response for porous functionally graded curved nanobeam can be obtained by solving Eq. (19) (B0 = ran = 0). While, for q0 = ran = 0, the buckling load can be obtained by solving Eq. (19) |[P I = 0. Also, for q0 = B0 = 0 the vibration can be obtained by solving Eq. (19) |[P | = 0.

4. Discussions

Numerical results are presented here to discuss the bending, buckling and vibration behaviours of porous functionally graded curved nanobeam with different boundary conditions. The present porous functionally graded curved nanobeam is made of the material properties which is given for steel as: Em = 210 GPa, pm = 7800 kg/m3 while for alumina Ec = 390 GPa, pc = 3960 kg/m3, v = 0.3. For the present results n = 1, L = 10 nm and the following dimensionless are used:

_ 102 Ech3 w = - c

12q0 L4

Wr, aee = -

2 L

10q0

Oft, = —

2 L

10q0

>ez>

2 ,P2h ^ 12R2

ra = L J-*-^-ra, r = D

h3 Ec

Eh3

B0, D =-L = Rp.

0 12(1 -v2)

Table 1 depicts the comparison with Ebrahimi and Barati [37] for the buckling load of perfect and porous functionally graded curved nanobeams for different (21) parameters, which illustrates consistency with our rede, sults. The inclusion of the porosity factor a is involved in this table.

Table 2 illustrates the nonlocal radial displacement W at the S-S end of the porous functionally graded curved nanobeam under different porosity factors,

Table 2. Nondimensional nonlocal radial displacement w at the S-S end of the porous functionally graded curved nanobeam under different porosity factor, power-law index, arc angle and nonlocal parameters (L/h = 10, £ = 2)

y a Local (^ = 0) Nonlocal (^ = 2)

P = rc/6 P = rc/4 P = rc/2 P = 2n/3 P = rc/6 P = rc/4 P = rc/2 P = 2n/3

1 0.0 1.25249 1.35078 2.12978 3.90697 1.49972 1.61741 2.55018 4.67818

0.1 1.40237 1.51284 2.38729 4.38175 1.67918 1.81147 2.85853 5.24667

0.2 1.59468 1.72091 2.71844 4.99293 1.90946 2.06061 3.25504 5.97850

2 0.0 1.34279 1.44859 2.28601 4.19608 1.60786 1.73453 2.73726 5.02435

0.1 1.51812 1.63829 2.58808 4.75378 1.81778 1.96169 3.09894 5.69213

0.2 1.74898 1.88829 2.98697 5.49124 2.09422 2.26102 3.57657 6.57517

3 0.0 1.38253 1.49126 2.35250 4.31716 1.65543 1.78562 2.81687 5.16934

0.1 1.56845 1.69241 2.67265 4.90810 1.87805 2.02648 3.20021 5.87692

0.2 1.81522 1.95959 3.09886 5.69598 2.17353 2.34640 3.71056 6.82032

5 0.0 1.42995 1.54177 2.42924 4.45449 1.71221 1.84611 2.90875 5.33377

0.1 1.62756 1.75541 2.76852 5.07987 1.94883 2.10191 3.31500 6.08260

0.2 1.89119 2.04063 3.22247 5.91782 2.26449 2.44343 3.85856 7.08595

10 0.0 1.50351 1.61990 2.54679 4.66336 1.80029 1.93966 3.04951 5.58387

0.1 1.72004 1.85363 2.91632 5.34251 2.05956 2.21952 3.49197 6.39707

0.2 2.01096 2.16784 3.41393 6.25810 2.40790 2.59575 4.08781 7.49340

power-law index, arc angle and nonlocal parameters. It can be noted that the local and nonlocal deflection increases as the power-law index y and arc angle P increase. As it is seen, the normalized S-S end nonlocal radial displacement w is greater than the local radial displacement w. Furthermore, there is an increase in the outcomes for higher porosity factor a. Table 3 illustrates the variation of radial displacement w of and stresses in perfect and porous functionally graded curved nanobeams under various boundary conditions. One can be noted that the normalized non-

7.4 -i a = 0.1 y — 1 l£.

7.1- i --y = 2

,^6.8-lfi.5- <D O a a 6.2- i*l H 5.9- K ---y = 3 ,v\ ----y = 4 w V

5.6- \ '---- _ _

5.3-

5.0- 1 1 1 1 1 1 1

4 6 8 10 12 14 16 18 20 Slenderness ratio L/h

Fig. 2. Variation of nonlocal deflection versus the slenderness nanobeams for different y (a) and a (b), ^ = £ = 2, P = 2n/3

local radial displacement w and stresses for porous functionally graded curved nanobeam are greater than those for nonlocal perfect functionally graded curved nanobeam. The largest values for nonlocal radial displacement w and stresses for perfect and porous functionally graded curved nanobeams occurred at S-S end but the smallest values occurred at F-F end.

Figure 2 demonstrates the variation of nonlocal radial displacement w versus slenderness ratio L/h at the S-S end of the porous functionally graded curved nanobeams for different y and a. It can be noted that

Slenderness ratio L/h

io L/h at the S-S end of the porous functionally graded curved

Table 3. Nondimensional radial displacement and stresses of perfect and porous functionally graded curved nanobeams under various boundary conditions (L/h = 10, P = rc/4, y = 2)

B.C 4 Perfect (a = 0) Porous (a = 0.2)

w ^ez w 0.^ee ^ez

S-S 0 0 1.73453 3.22829 8.02455 2.26102 3.42922 7.95646

1 1.90573 3.89698 9.68669 2.48417 4.13952 9.60451

2 2.07692 4.62856 11.50517 2.70733 4.91663 11.40756

1 0 1.57872 2.93829 7.30369 2.05791 3.12117 7.24173

1 1.73453 3.54691 8.81654 2.26102 3.76767 8.74173

2 1.89035 4.21277 10.47166 2.46413 4.47497 10.38282

C-S 0 0 0.23544 1.27687 5.45225 0.30503 1.36434 5.37419

1 0.25868 1.64274 7.46843 0.33514 1.75232 7.36151

2 0.28192 2.05169 9.75016 0.36524 2.18582 9.61057

1 0 0.18789 1.05473 4.64824 0.24353 1.12859 4.58401

1 0.20643 1.35992 6.36710 0.26757 1.45289 6.27912

2 0.22497 1.70124 8.31236 0.29160 1.81545 8.19749

C-C 0 0 0.15231 0.56267 2.74603 0.19707 0.60493 2.70388

1 0.16734 0.74000 4.20814 0.21652 0.79318 4.14355

2 0.18237 0.93922 5.88423 0.23597 1.00452 5.79392

1 0 0.12806 0.47308 2.30883 0.16588 0.50918 2.27589

1 0.14069 0.62219 3.53815 0.18225 0.66763 3.48768

2 0.15334 0.78969 4.94739 0.19862 0.84551 4.87682

F-F 0 0 0.11779 0.63282 4.03956 0.15244 0.68283 3.97973

1 0.12941 0.87682 7.24169 0.16748 0.94197 7.13444

2 0.14104 1.15344 10.94749 0.18253 1.23557 10.78535

1 0 0.07809 0.42593 3.16713 0.10125 0.46070 3.12620

1 0.08581 0.59275 5.67769 0.11124 0.63851 5.60432

2 0.93515 0.78197 8.58314 0.12123 0.84009 8.47222

the radial displacement w decreases as the slender-ness ratio L/h increases. It is obvious from Fig. 2 that the nonlocal radial displacement w increases with the rise of y and a for all the ratio L/h. The variation of nonlocal stresses cee, cez through the side-to-thickness ratio L/h at the S-S end of the porous functionally graded curved nanobeams for different ^ are plotted in Fig. 3. It can be concluded, the nonlocal stress Cee increases through the side-to-thickness ratio L/h with the growth of On the contrary, the nonlocal stress cez decreases along the side-to-thickness ratio L/h of the porous functionally graded curved nanobeam with the growth of The effect of arc angle p and nonlocal parameter ^ can be examined simultaneously in one graph. The variation of nonlocal stresses Cee, Cez through the arc angle p at the S-S end of the porous functionally graded curved nanobeams for dif-

ferent ^ are plotted in Fig. 4. It is obvious that the nonlocal stress cee decreases with the rise of ^ for all arc angles 6 <p< 4rc/ 9 but it increases with the rise of ^ for all arc angles 4rc/9 < p < 2rc/3. While the nonlocal stress cez decreases with the rise of ^ for all arc angles 6 <p< 2rc/ 3.

Figure 5 demonstrates the variation of nonlocal stress cee through the thickness of the porous functionally graded curved nanobeams for different y and a. It is obvious that the nonlocal stress cee decreases with the rise of y in the interval -0.5 < z/h < 0.2 but it increases with the rise of y in the interval 0.2 < z/h < 0.5. However, the nonlocal stress cee decreases with the rise of porosity factor a in the interval -0.5 < z/h < 0.05 but it increases with the rise of porosity factor a in the interval 0.05 < z/h < 0.5 at the S-S end for the porous functionally graded curved nanobeam as given

Fig. 3. Variation of nonlocal stresses aee (a) and a0z (b) versus the slenderness ratio L/h at the S-S end of the porous functionally graded curved nanobeams for different ц, ß = 2л/3, a = 0.1, у = 1, £ = 2

Fig. 4. Variation of nonlocal stresses aee (a) and aez (b) versus the arc angle ß at the S-S end of the porous functionally graded curved nanobeams for different ц, £ = 2, a = 0.1, у = 2, L/h = 10

Fig. 5. Variation of nonlocal stress aee through the thickness at the S-S end of the porous functionally graded curved nanobeams for different у (a) and a (b), ß = 2л/3, L/h = 10, £ = ц = 2

z/h 0.4 H

0.2-

o.o-

-0.2-

-0.4-

l£

y = 0.0 V N • \

--y = 0.1 )>• \

---y = 0.5 /

----y= 1.0 A ''y

V *

*

0

-1 i 4

12

16 CJ0z

z/h' 0.4

0.2

0.0

-0.2

-0.4

lb.

a = 0.0

--a = 0.1 \\\ \

---a = 0.2 1 / \

----a = 0.3 /// ^ * /

0

12

16 cj0z

Fig. 6. Variation of nonlocal stress a0z through the thickness at the S-S end of the porous functionally graded curved nanobeams for different y (a) and a (b), P = 2n/3, L/h = 10, £ = ^ = 2

in Fig. 5. Figure 6 illustrates the variation of nonlocal stress a0z through-the-thickness of the porous functionally graded curved nanobeam for different power-law index y and porosity factor a. It is found that the nonlocal stress a0z increases in all the length of the porous functionally graded curved nanobeam with the growth of power-law index y and porosity factor a.

Table 4 illustrates the effects of porosity factor, various opening angles, power-law index on the buckling for various boundary conditions of porous functionally graded curved nanobeam (L/h = 10, ^ = £ = 2). One can notice that for the two parameters y, a, the values diffusion r decrease with the increase of the curved angle p. The inclusion of various boundary

conditions is involved in Table 4. Table 5 demonstrates the variation of the vibration of S-S porous functionally graded curved nanobeam with different power-law index, porosity factor, length scale and the angle P (L/h = 10, ^ = 2). It can be noted that for the three parameters y, a, P, the vibration values decrease with the increase in the strain gradient parameter £. Table 6 demonstrates the effects of nonlocal parameter, various boundary conditions, porosity factor, various opening angles on the vibration of porous functionally graded curved nanobeam (L/h = 10, y = £ = 2). One can note that for the two parameters a, the vibration values decrease with the increase of the curved angle P. One can also note that the smallest value for

Table 4. Effects of porosity factor, various opening angles, power-law index, boundary conditions on the buckling load of perfect and porous functionally graded curved nanobeams (L/h = 10, ^ = £ = 2)

B.C a p = rc/4 p = rc/2

y = 0 y il 5 y = 10 y = 0 y il 5 y = 10

S-S 0.0 13.7044 8.8974 8.4683 2.1894 1.4117 1.3466

0.1 12.6502 7.8146 7.4005 2.0210 1.2387 1.1759

0.2 11.5961 6.7223 6.3279 1.8526 1.0642 1.0045

C-S 0.0 63.3903 40.2419 38.1811 32.1299 20.0028 18.9258

0.1 58.5141 35.2845 33.2652 29.6583 17.5137 16.4455

0.2 53.6379 30.3008 28.3353 27.1868 15.0189 13.9622

C-C 0.0 90.2025 57.1434 54.2174 46.7038 29.0590 27.4956

0.1 83.2638 50.0923 47.2228 43.1112 25.4409 23.8903

0.2 76.3252 43.0049 40.2084 39.5186 21.8148 20.2804

F-F 0.0 150.2091 94.9726 90.2155 67.1602 41.8399 39.6424

0.1 138.6546 83.2133 78.5533 61.9940 36.6252 34.4505

0.2 127.1000 71.3889 66.8537 56.8279 31.3950 29.2496

Table 5. Non-dimensional of vibration of S-S perfect and porous functionally graded curved nanobeams for various power-law index, porosity factor, opening angles and length scale parameters (L/h = 10, | = 2)

y a 4 = 0 4 = 1

P = ft/4 P = ft/3 P = ft/2 P = 2 ft/3 P = ft/4 P = ft/3 P = ft/2 P = 2 ft/3

0 0.0 7.69545 7.13432 5.67475 3.91065 8.06627 7.47809 5.94819 4.09909

0.1 8.01229 7.42806 5.90839 4.07166 8.39838 7.78599 6.19310 4.26787

0.2 8.44251 7.82691 6.22564 4.29029 8.84933 8.20406 6.52564 4.49703

1 0.0 5.39699 4.99033 3.95602 2.72350 5.65706 5.23080 4.14665 2.85474

0.1 5.37109 4.96499 3.93455 2.70845 5.62991 5.20425 4.12414 2.83897

0.2 5.33581 4.93067 3.90565 2.68824 5.59293 5.16827 4.09385 2.81778

2 0.0 4.94965 4.57591 3.62625 2.49565 5.18816 4.79641 3.80099 2.61591

0.1 4.87538 4.50582 3.56919 2.45597 5.11031 4.72294 3.74117 2.57432

0.2 4.77941 4.41536 3.49566 2.40486 5.00972 4.62812 3.66410 2.52075

5 0.0 4.58776 4.24384 3.36536 2.31624 4.80883 4.44834 3.52752 2.42785

0.1 4.48489 4.14752 3.28767 2.26234 4.70099 4.34737 3.44609 2.37136

0.2 4.35561 4.02649 3.19009 2.19465 4.56549 4.22052 3.34381 2.30039

10 0.0 4.39567 4.06905 3.22978 2.22367 4.60749 4.26512 3.38542 2.33083

0.1 4.27979 3.96097 3.14307 2.16365 4.48603 4.15184 3.29452 2.26791

0.2 4.13643 3.82725 3.03576 2.08935 4.33575 4.01167 3.18204 2.19003

the vibration occurred at S-S boundary conditions and the largest value occurred at F-F boundary conditions.

Figure 7 demonstrates the effect of porosity factor a, nonlocal parameter | and power-law index y on the buckling load of porous functionally graded curved nanobeams with S-S boundary condition (4 = 2, L/h = 10, p = ft/2). Figure 7, a indicates that the buckling decreases as a and | decrease for fixed y = 2, while, Fig. 7, b points out that the buckling load decreases as

y and | increases for fixed a = 0.1. The differences in buckling may be more than double for greater values of the parameter a. Figure 8 demonstrates the effect of the strain gradient parameter 4, the parameter a and power-law index y on the buckling of porous functionally graded curved nanobeams with S-S boundary condition (| = y = 2, L/h = 10, p = ft/2, a = 0.1). Figure 8, a displays that the buckling increases as the parameter 4 increases for fixed y = 2, while Fig. 8, b shows

Table 6. Effects of boundary conditions, porosity factor, various opening angles, nonlocal parameter on the vibration of perfect and porous functionally graded curved nanobeams (L/h = 10, y = 4 = 2)

B.C a P = ft/6 P = 2 ft/3

| = 0 1 = 1 | = 2 | = 0 1 = 1 | = 2

S-S 0.0 6.27081 5.98253 5.73067 2.98827 2.85089 2.73087

0.1 6.17907 5.89501 5.64683 2.94076 2.80557 2.68746

0.2 6.06035 5.78175 5.53834 2.87956 2.74719 2.63153

C-S 0.0 16.71164 15.51637 14.54559 31.09931 28.77028 26.89621

0.1 16.47918 15.30002 14.34242 30.69118 28.39059 26.53978

0.2 16.18078 15.02231 14.08162 30.18325 27.91812 26.09626

C-C 0.0 19.89491 18.66949 17.64572 38.75145 36.17726 34.05338

0.1 19.62203 18.41223 17.40166 38.25579 35.70876 33.60813

0.2 19.27203 18.08224 17.08861 37.63833 35.12519 33.05357

F-F 0.0 28.25947 26.67465 25.32953 49.05882 46.1876 43.76505

0.1 27.87332 26.30841 24.98040 48.43644 45.59443 43.19737

0.2 27.37826 25.83879 24.53268 47.65799 44.85243 42.48724

0123ц 0 12 3ц

Fig. 7. Influence of the nonlocal parameter ц, porosity factor a (a) and power-law index y (b) on the buckling of S-S porous functionally graded curved nanobeams, £ = 2, L/h = 10, ß = я/2

0 1 2 3 £ 0 1 2 3 £

Fig. 8. Influence of the strain gradient parameter £, porosity factor a (a) and power-law index y (b) on the buckling of S-S porous functionally graded curved nanobeams, ц = 2, L/h = 10, ß = я/2

4 8 12 16 Llh 4 8 12 16 Llh

Fig. 9. Influence of the porosity factor a (a) and power-law index y (b) on the buckling of S-S porous functionally graded curved nanobeams through the side-to-thickness ratio L/h, ц = £ = 2, L/h = 10, ß = я/2

Fig. 10. Influence of the nonlocal parameter ц on the vibration of S-S porous functionally graded curved nanobeams for different values of porosity factor a (a) and power-law index y (b), L/h = 10, ß = я/4, £ = 2

Fig. 11. Influence of the strain gradient parameter £ on the vibration of S-S porous functionally graded curved nanobeams for different values of porosity factor a (a) and power-law index y (b), L/h = 10, ß = я/4, ц = 2

Fig. 12. Influence of the nonlocal parameter ц versus the slenderness ratio L/h (a) and strain gradient parameter £ (b) on the vibration of S-S porous functionally graded curved nanobeams, a = 0.1, ß = я/4, y = 2

that the buckling decreases as y increases for fixed a = 0.1. Figure 9 shows the effect of porosity factor a and the power-law index y on the buckling of porous functionally graded curved nanobeam through the side-to-thickness ratio L/h with S-S boundary condition (| = 4 = 2, L/h = 10, P = ft/2). It can be noted that the buckling increases as the side-to-thickness ratio L/h increases while, it decreases as a, y increase for fixed y = 2 in Fig. 9, a and for fixed a = 0.1 in Fig. 9, b.

Figure 10 shows the effect of the nonloacl parameter | on the vibration of S-S porous functionally graded curved nanobeams for different values of the porosity factor a (Fig. 10, a) and the power-law index y (Fig. 10, b) (L/h = 10, P = ft/4, 4 = 2). Because of the increase in the three parameters (a, y, |) the vibrations decrease. Figure 11 illustrates the effect of strain gradient parameter 4 on the vibration of S-S porous functionally graded curved nanobeams for various values of porosity factor a (Fig. 11, a) and the power-law index y (Fig. 11, b) (L/h = 10, P = ft/4, | = 2). Figure 12 demonstrates the effect of the nonlocal parameter | through the side-to-thickness ratio L/h and the strain gradient parameter 4 on the vibration of S-S porous functionally graded curved nanobeam (a = 0.1, P = ft/4, y = 2). It can be noted that the vibration increases by an increase in the ratio L/h.

5. Conclusions

This paper demonstrates the bending, buckling and free vibration on porous functionally graded curved nanobeams under different boundary conditions via a nonlocal strain gradient theory. The size impacts are considered by employing the nonlocal strain gradient theory that contain both the nonlocal and gradient coefficients. These coefficients have different physical effects as shown in the numerical outcomes. The dynamic equations are deduced according to the present beam model. Comparison example is presented, which reveals that the present results for a perfect functionally graded curved nanobeam have large agreements with the previously published outcomes. Moreover, several numerical examples are performed to show the impacts of the porosity factor, nonlocal and gradient parameters, power-law index, beam angle, side-to-thickness ratio and different boundary conditions on the buckling and free vibration in the porous functionally graded curved nanobeams. So, the present results are very consistent with those given in the literature. The buckling loads of the porous functionally graded curved nanobeams have significant changes with the

beam angle and the porosity factor. The buckling loads and vibration frequencies for porous and non-porous functionally graded curved nanobeams are generally lower than the corresponding values for homogeneous ceramic nanobeams. The results show that the presence of porosity factor leads to a significant increment in the buckling loads and the variation frequencies.

References

1. Hosseini-Hashemi S., Nahas I., Fakher M., Nazemnez-had R. Surface effects on free vibration of piezoelectric functionally graded nanobeams using nonlocal elasticity // Acta Mech. - 2014. - V. 225. - P. 1555-1564.

2. Niknam H., Aghdam M. A semi-analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation // Compos. Struct. - 2015. - V. 119. - P. 452-462.

3. Rahmani O., Pedram O. Analysis and modeling the size effect on vibration of functionally graded nano-beams based on nonlocal Timoshenko beam theory // Int. J. Eng. Sci. - 2014. - V. 77. - P. 55-70.

4. Ebrahimi F., Salari E. Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment // Acta Astronaut. - 2015. -V. 113. - P. 29-50.

5. Reddy J.N. Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates // Int. J. Eng. Sci. - 2010. - V. 48. -P. 1507-1518.

6. Reddy J.N. Nonlocal theories for bending, buckling and vibration of beams // Int. J. Eng. Sci. - 2007. - V. 45. -P. 288-307.

7. Karami B., Janghorban M., Tounsi A. Nonlocal strain gradient 3D elasticity theory for anisotropic spherical na-noparticles // Steel Compos. Struct. - 2018. - V. 27. -P. 201-216.

8. Radwan A.F., Sobhy M. A nonlocal strain gradient model for dynamic deformation of orthotropic viscoelastic graphene sheets under time harmonic thermal load // Physica B. - 2018. - V. 538. - P. 74-84.

9. Barati M.R., Zenkour A.M. Post-buckling analysis of imperfect multiphase nanocrystalline nanobeams considering nanograins and nanopores surface effects // Compos. Struct. - 2018. - V. 184. - P. 497-505.

10. Roque C.M.C., Ferreira A.J.M., Reddy J.N. Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method // Int. J. Eng. Sci. - 2011. -V. 49. - P. 976-984.

11. Thai H.-T. A nonlocal beam theory for bending, buckling, and vibration of nanobeams // Int. J. Eng. Sci. -2012. - V. 52. - P. 56-64.

12. Ansari R., Sahmani S. Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories // Int. J. Eng. Sci. -2011. - V. 49. - P. 1244-1255.

13. Gheshlaghi B., Hasheminejad S. Surface effects on nonlinear free vibration of nanobeams // Compos. B. -2011. - V. 42. - P. 934-937.

14. Phadikar J.K., Pradhan S.C. Variational formulation and finite element analysis for nonlocal elastic nano-beams and nanoplates // Comput. Mater. Sci. - 2010. -V. 49. - P. 492-499.

15. Foroushani S.S., Azhari M. Nonlocal vibration and buckling analysis of single and multi-layered graphene sheets using finite strip method including van der Waals effects // Physica E. - 2014. - V. 57. - P. 83-95.

16. Barati M.R., Zenkour A. A general bi-Helmholtz nonlocal strain-gradient elasticity for wave propagation in nanoporous graded double-nanobeam systems on elastic substrate // Compos. Struct. - 2017. - V. 168. -P. 885-892.

17. Aydogdu M. A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration // Physica E. - 2009. - V. 41. - P. 1651-1655.

18. Zenkour A.M. Nonlocal elasticity and shear deformation effects on thermal buckling of a CNT embedded in a viscoelastic medium // Eur. Phys. J. Plus. - 2018. -V. 133. - P. 1-14.

19. Zenkour A.M., Aljadani M.H. Mechanical buckling of functionally graded plates using a refined higher-order shear and normal deformation plate theory // Adv. Aircraft Spacecraft Sci. - 2018. - V. 5. - P. 615-632.

20. Bouazza M., Benseddiq N., Zenkour A.M. Thermal buckling analysis of laminated composite beams using hyperbolic refined shear deformation theory // J. Therm. Stresses. - 2019. - V. 42. - P. 332-340.

21. Zenkour A.M., Aljadani M.H. Thermo-electrical buckling response of actuated functionally graded piezoelectric nanoscale plates // Results Phys. - 2019. - V. 13. -P. 102192.

22. Bouazza M., Zenkour A.M., Benseddiq N. Closed-from solutions for thermal buckling analyses of advanced na-noplates according to a hyperbolic four-variable refined theory with small-scale effects // Acta Mech. - 2018. -V. 229. - P. 2251-2265.

23. Zenkour A.M. Buckling of a single-layered graphene sheet embedded in visco-Pasternak's medium via nonlocal first-order theory // Adv. Nano Res. - 2016. -V. 4. - P. 309-326.

24. Li L., Hu Y. Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory // Int. J. Eng. Sci. - 2015. - V. 97. - P. 84-94.

25. Li L., Li X., Hu Y. Free vibration analysis of nonlocal strain gradient beams made of functionally graded material // Int. J. Eng. Sci. - 2016. - V. 102. - P. 77-92.

26. Simsek M. Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach // Int. J. Eng. Sci. -2016. - V. 105. - P. 12-27.

27. Allam M.N.M., Radwan A.F. Nonlocal strain gradient theory for bending, buckling, and vibration of viscoe-lastic functionally graded curved nanobeam embedded

in an elastic medium // Adv. Mech. Eng. - 2019. -V. 11. - P. 1687814019837067.

28. Yang F., Chong A.C.M., Lam D.C.C., Tong P. Couple stress based strain gradient theory for elasticity // Int. J. Solids Struct. - 2002. - V. 39. - P. 2731-2743.

29. Eringen A.C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves // J. Appl. Phys. - 1983. - V. 54. - P. 4703-4710.

30. Aifantis E.C. Strain gradient interpretation of size effects // Int. J. Fract. - 1999. - V. 95. - P. 1-4.

31. Eringen A.C. Theory of micropolar plates // Z. Angew. Math. Phys. - 1967. - V. 18. - P. 12-30.

32. Eringen A.C. Nonlocal polar elastic continua // Int. J. Eng. Sci. - 1972. - V. 10. - P. 1-16.

33. Eringen A.C., Edelen D.G.B. On nonlocal elasticity // Int. J. Eng. Sci. - 1972. - V. 10. - P. 233-248.

34. Eringen A.C. Nonlocal Continuum Field Theories. -New York: Springer-Verlag, 2002.

35. Rezaiee-Pajand M., Rajabzadeh-Safaei N. Nonlocal static analysis of a functionally graded material curved nanobeam // Mech. Adv. Mater. Struct. - 2018. -V. 25. - P. 539-547.

36. Arefi M., Zenkour A.M. Electro magneto elastic analysis of a three layer curved beam // Smart Struct. Systems. - 2017. - V. 19. - P. 695-703.

37. Ebrahimi F., Barati M.R. A nonlocal strain gradiant refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams // Compos. Struct. - 2017. - V. 159. - P. 174-182.

38. Medina L., GilatR., Krylov S. Symmetry breaking in an initially curved micro beam loaded by a distributed electrostatic force // Int. J. Solids Struct. - 2012. -V. 49. - P. 1864-1876.

39. Filipich C.P., PiovanM.T. The dynamics of thick curved beams constructed with functionally graded materials // Mech. Res. Commun. - 2010. - V. 37. - P. 565-570.

40. Assadi A., Farshi B. Size dependent vibration of curved nanobeams and rings including surface energies // Phy-sica E. - 2011. - V. 43. - P. 975-978.

41. Barati M.R., Zenkour A.M. Post-buckling analysis of refined shear deformable graphene platelet reinforced beams with porosities and geometrical imperfection // Compos. Struct. - 2017. - V. 181. - P. 194-202.

42. Barati M.R., Zenkour A.M. Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions // Compos. Struct. - 2017. - V. 182. - P. 91-98.

43. Barati M.R., Zenkour A.M. Analysis of postbuckling of graded porous GPL-reinforced beams with geometrical imperfection // Mech. Adv. Mater. Struct. - 2019. -V. 26. - P. 503-511.

44. Barati M.R., Zenkour A.M. Analysis of postbuckling behavior of general higher-order functionally graded nanoplates with geometrical imperfection considering porosity distributions // Mech. Adv. Mater. Struct. -2019. - V. 26. - P. 1081-1088.

45. Behravan R.A., Shariyat M. Three-dimensional magneto-elastic analysis of asymmetric variable thickness porous FGM circular plates with non-uniform tractions and Kerr elastic foundations // Compos. Struct. -2015. - V. 125. - P. 558-574.

46. Al Rjoub Y.S., Hamad A.G. Free vibration of functionally Euler Bernoulli and Timoshenko graded porous beams using the transfer matrix method // KSCE J. Civ. Eng. - 2017. - V. 21. - P. 792-806.

47. Ghorbanpour A.A., Khani M., Khoddami M.Z. Dynamic analysis of a rectangular porous plate resting on an elastic foundation using high-order shear deformation theory // J. Vib. Control. - 2017. - P. 1-16.

48. Wu D., Liu A., Huang Y., Huang Y., Pi Y., Gao W. Dynamic analysis of functionally graded porous structures through finite element analysis // Eng. Struct. - 2018. -V. 165. - P. 287-301.

49. Li L., Tang H., Hu Y. Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature // Compos. Struct. - 2018. -V. 184. - P. 1177-1188.

50. Jabbari M., Mojahedin A., Haghi M. Buckling analysis of thin circular FG plates made of saturated porous-soft ferromagnetic materials in transverse magnetic field // Thin-Walled Struct. - 2014. - V. 85. - P. 50-56.

51. Cong P.H., Chien T.M., Khoa N.D., Duc N.D. Nonlinear thermomechanical buckling and post-buckling response of porous FGM plates using Reddy's HSDT // Aerosp. Sci. Technol. - 2018. - V. 77. - P. 419-428.

52. Feyzi M.R., Khorshidvand A.R. Axisymmetric post-buckling behavior of saturated porous circular plates // Thin-Walled Struct. - 2017. - V. 112. - P. 149-158.

53. Rezaei A.S., Saidi A.R. Buckling response of moderately thick fluid-infiltrated porous annular sector plates // Acta Mech. - 2017. - V. 228. - P. 3929-3945.

54. Shojaeefard M.H., Googarchin H.S., Ghadiri M., Ma-hinzare M. Micro temperature dependent FG porous plate: free vibration and thermal buck-ling analysis using modified couple stress theory with CPT and FSDT // Appl. Math. Model. - 2017. - V. 50. - P. 633-655.

55. Yang J., Chen D., Kitipornchai S. Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Cheby-shev-Ritz method // Compos. Struct. - 2018. -V. 193. - P. 281-294.

56. Chen D., Yang J., Kitipornchai S. Nonlinear vibration and postbuckling of functionally graded graphene reinforced porous nanocomposite beams // Compos. Sci. Technol. - 2017. - V. 142. - P. 235-245.

57. Radwan A.F. Quasi-3D integral model for thermomechanical buckling and vibration of FG porous nano-plates embedded in an elastic medium // Int. J. Mech. Sci. - 2019. - V. 157. - P. 320-335.

58. AifantisE.C. On the gradient approach-relation to Erin-gen's nonlocal theory // Int. J. Eng. Sci. - 2011. - 49. -P. 1367-1377.

59. Lim C.W., Zhang G., Reddy J.N. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation // J. Mech. Phys. Solids. -2015. - V. 78. - P. 298-313.

60. Zenkour A.M., Sobhy M. Nonlocal piezo-hygrothermal analysis for vibration characteristics of a piezoelectric Kelvin-Voigt viscoelastic nanoplate embedded in a vis-coelastic medium // Acta Mech. - 2018. - V. 229. -P. 3-19.

Received 29.04.2020, revised 15.06.2020, accepted 15.06.2020

Сведения об авторах

Ashraf M. Zenkour, PhD, Prof., King Abdulaziz University, Saudi Arabia, [email protected], [email protected]

Ahmed F. Radwan, PhD, Ass. Prof., Higher Institute of Management and Information Technology, Nile for Science and Technology,

Egypt, [email protected]