Extended four-unknown higher-order shear deformation nonlocal theory for bending, buckling and free vibration of functionally graded porous nanoshell resting on elastic foundation

Extended four-unknown higher-order shear deformation nonlocal theory for bending, buckling and free vibration of functionally graded porous nanoshell resting on elastic foundation

Trung Thanh Tran

^a

, Van Ke Tran

^a

, Quoc-Hoa Pham

^b,c

, Ashraf M. Zenkour

^d,e,⇑

aDepartment of Mechanics, Le Quy Don Technical University, Hanoi, Viet Nam

bDivision of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

cFaculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

dDepartment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

eDepartment of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt

A R T I C L E I N F O

Keywords:

Nonlocal theory Functionally graded porous Elastic foundation Nanoshell

A B S T R A C T

This article aims to study bending, buckling, and free vibration behaviors of the functionally graded porous (FGP) nanoshell resting on an elastic foundation (EF) including static bending, free vibration, hydro‐

thermal–mechanical buckling. We use the four‐unknown high‐order shear deformation theory based on Eringen’s nonlocal theory and Hamilton’s principle to obtain the system of the governing differential equations.

By using Navier's solution, the static, buckling, and free vibration responses of the FGP nanoshells are solved.

The FGP material with uneven porosity and logarithmic‐uneven porosity distribution is employed. The EF is a Winkler‐Pasternak foundation with the stiffness coefficientk_wand sliding stiffness coefficientk_s. The numerical results in the present work are compared with those of the published works to evaluate the accuracy and reli- ability of the proposed formulas. Afterward, the influences of the geometric dimensions, material properties, and the elastic foundation stiffness on the response of the FGP nanoshell is studied in detail.

1. Introduction

Nowadays, nanotechnology has achieved great achievements in manyfields such as electronics, biomedical, aerospace, etc. The inves- tigation of nanostructures has always been deeply concerned by scien- tists in the world include mechanical research. However, studies show that conventional computational theories for structures of millimeters or above are inaccurate for nanometer‐sized structures. The first moment, molecular dynamic simulation in [1–3] has also included the effect of length‐sized. However, this theory involves solving a large number of equations. Therefore, they are inefficient and expensive computation cost. Along with the development of the calculation method, a lot of theories for calculating nanostructures have been pro- posed such as the modified couple stress theory[4], the strain gradient theory[5], the nonlocal theory[6–9]. As it is known, the nonlocal the- ory is preferred and used widely by researchers for its simplicity and accuracy. The basic problem of this theory is based on the assumption that stress at one point is a continuum function of the strain at all neighboring points.

Using the nonlocal theory to investigate mechanical behaviors of nanostructures: Sobhy and his colleagues[10]used a four‐unknown and quasi‐3D higher‐order shear deformation theories to study the sta- tic bending, free vibration, mechanical buckling, and thermal buckling of FGM nanoplates on EF. Calculating the static and dynamic of FGM plates using a four‐variable refined plate theory is presented by Mechab[11]. Shobhy et al.[12–16]developed the quasi‐3D refined plate theory to analyze the mechanical behavior of FGP structures as static bending, free vibration, and buckling. Wang et al.[17]investi- gated the effect of a small length‐scale on the static bending of nano- plates resting on EF using an analytical method (AM). Rouabhia et al.

[18]also employed the nonlocal theory to analyze the buckling of an SLGS placed on a viscoelastic medium. Panyatong [19] based on Navier’s solution to study the effects of surface stress on the static bending of nanoplates. Reddy et al.[20]used third‐order shear defor- mation theory (TSDT) to solve the static bending and free vibration of nanoplates. Besides, Panyatong et al.[21]based on the second‐order shear deformation theory (SSDT) to calculate the free vibration of FGM nanoplates. We also can find valuable results for the free

https://doi.org/10.1016/j.compstruct.2021.113737

Received 8 December 2020; Revised 10 February 2021; Accepted 11 February 2021 Available online 18 February 2021

0263-8223/© 2021 Elsevier Ltd. All rights reserved.

⇑Corresponding author at: Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.

E-mail addresses:phamquochoa@tdtu.edu.vn(Q.-H. Pham),zenkour@kau.edu.sa,zenkour@sci.kfs.edu.eg(A.M. Zenkour).

Composite Structures 264 (2021) 113737

Contents lists available atScienceDirect

Composite Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c t

vibration analysis of double‐nanoplate‐systems[22]and single‐layered nano‐graphene sheet[23]. Ansari et al.[24]studied the free vibrations of single‐layered graphene sheets using the nonlocal theory. In addi- tion, Hashemi et al.[25]proposed an exact analytical approach for the free vibration analysis of nanoplates, and Pouresmaeeli et al.

[26] also analyzed the free vibration of orthotropic nanoplates on EF. Wang et al.[27]used Navier’s solution based on the high‐order shear deformation theory (HSDT) to investigate the bending, free vibration of the FGM nanoshells reinforced by graphene nanoplatelets.

Shah et al.[28]examined the static bending of double‐curved shells.

Ke et al.[29]calculated the free vibration of magneto‐electro‐elastic (MEE) nanoplates using the AM. Viola et al. [30] studied the free vibration of double‐curved laminated shells/panels based on HSDT.

Salehipour et al.[31]employed an exact analytical solution for the free vibration analysis of FGM micros/nanoplates following three‐ dimensional nonlocal theory. Natarajan et al. [32] studied the free vibration of FGM nanoplates using the Mori–Tanaka homogenization scheme. Jung et al.[33]employed Navier’s solution to analyze the sta- tic bending, free vibration of sigmoid‐FGM nanoplates. Moreover, Kar- ami et al. [34] showed a lot of useful results about the dynamic response of FGP nanoshells and Mohammad et al.[35]indicated the dynamic stability of the FGP nanoplate via a nonlocal strain gradient based on the Quasi‐3D theory. In addition, Berghouti et al.[36]also employed the nonlocal theory to the examine free vibration of FGP nanobeams and so on. In thefield of buckling analysis of nanostruc- tures: Prandhan et al.[37]proposed a nonlocal theory to studied the buckling of nanoplates based on FSDT by using Navier's approach and investigated the effect of small scale‐length on the buckling of single‐layered graphene sheets[38]. Zenkour and Sobhy[39]exam- ined thermal buckling of nanoplates on the EF based on the sinusoidal shear deformation plate theory and the nonlocal theory. Khorshidi et al.[40]analyzed the buckling of FGM nanoplates using nonlocal exponential shear deformation theory. Tsai et al. [41] discussed a novel three‐dimensional deformation nonlocal theory to analyze FG magneto‐electro‐elastic nanoshells for static bending, free vibration, and buckling. Some different results about free vibration and buckling of FGM nanoshells are also presented in[42]. Shahsavari et al.[43]

developed a new size‐dependent quasi‐3D shear deformation theory to examine the buckling of FGP nanoplates.

Recently, a lot of analytical and numerical studies on the FGP struc- tures using different theories can be summarized as follows: Zine et al.

[44] analyzed static bending of FGP plates based on a cubic shear deformation theory using Navier’s solution. Bekkaye et al.[45]studied mechanical behaviors of FGM plates employed refined trigonometric shear deformation theory. Medani and co‐workers [46] examined the static and dynamic of FGP sandwich plates employed energy prin- ciple. Yahia et al.[47]investigated wave propagation in FGP plates based on using HSDT and Bennai et al.[48]employed a four variable plate theory. Cuong‐Le et al. [49] using IGA based on a three‐

dimensional solution to analyze the mechanical behavior of FGP struc- tures. Bennai and his colleagues[50]used the HSDT and the classical plate theory to investigate the free vibration of FGP plates. Tran et al.

[51]employed ES‐MITC3 element to examine vibration of FGP plates with variable thickness. Moreover, numerical results of mechanical behaviors of FGP beams can be found that in refs.[52–54]. For the investigation of FGP structures resting on EF: Kaddari et al.[55]novel a new quasi‐3D theory to compute bending and free vibration of FGP plates while Addou and co‐workers [56] analyzed the dynamic response of FGP plates employed an HSDT quasi‐3D theory. Nebab et al.[57,58]developed refined shear deformation theory to investi- gate the static and dynamic behaviors of FGM plates using Navier’s solution. Tran and co‐workers analyzed free vibration[59]and force vibration[60]of FGP plates resting on elastic foundation taking into mass (EFTIM) using the smooth‐FEM. Tounsi et al.[61]employed a

four‐unknown plate theory for hygro‐thermo‐mechanical bending analysis of AFG ceramic–metal plates. Furthermore, the valuable results on structural analysis on the EF can be found in Ref.[62].

Recent researches on the structures in thermal environments have attracted great interest from many researchers. The valuable results on structural analysis on the EF can be found in[63–75]. Mechab et al.[76]investigated composite plates used the HSDT subjected to the thermo‐mechanical load. Zhao et al.[77]studied the effect of ther- moelastic on the free vibration of FG cylindrical shells using Ritz’s method and employing Laplace–Fourier transformation by Kiani et al.[78]. Pradyumna[79]also analyzed the free vibration and buck- ling of FGM shells/panels based on HSDT. Tran and his colleagues pre- sented the prediction of the natural frequencies of FGM plates using an artificial neural network (ANN)[80]and calculated bending and buck- ling of unsymmetric FG sandwich beams based on a new TSDT[81].

Malekzadeh et al.[82]displayed the numerical results for the free vibration analysis of FGM cylindrical shells. Furthermore, Nami et al.[83]employed TSDT via nonlocal theory to calculate the thermal buckling of FGM nanoplates. Some results of the free vibration of orthotropic nanoplates in thermal environments are also shown in [84]. Reddy et al.[85]presented thermo‐electro‐mechanical vibration of nanoshells with different boundary conditions (BCs) using the dif- ferential quadrature method (DQM). Alghanmi and Zenkour[86]stud- ied the effect of porosity on the bending of functionally graded plates integrated with piezoelectricfiber‐reinforced composite layers. Fazel- zadeh et al.[87]based on the nonlocal theory to examine the free vibration of the single‐layered graphene sheet.

From the aforementioned literature reviews and according to the best knowledge of the authors, using the four‐unknown higher‐order shear deformation theory to study nanoshell is not published yet. In this work, we develop this theory for investigating the static bending, free vibration, and hydro‐thermal‐mechanical buckling of FGP nano- shells resting on the EF. The advantages of using the four‐unknown higher‐order shear deformation theory are improved accuracy, save time cost, exactly describe the stress–strainfield. Besides, the accuracy and reliability of the proposed method are performed by comparing the present numerical results with those of other published works.

Finally, the influences of the mechanical behavior of nanostructures, the effects of geometric parameters, material properties on the response of the FGP nanoshells are also examined in detail.

2. Theoretical formulation 2.1. FGP nanoshell resting on EF

We consider the FGP nanoshell resting on the EF (seeFig. 1) made from the mixture of ceramics and metals with the geometry parameter:

the curvature radius (R1,R2); the length of curve edge (a,b) and the thickness of nanoshellsh(0). The material properties are assumed that varying continuously from a top surface (z¼ þh=2) to the bottom (z¼ h=2) surface according to a power‐law distributionn. The elastic foundation is the Winkler‐Pasternak foundation consisting of two parameters: the stiffness coefficientkwand the sliding stiffness coeffi- cientks:

The law of changing the mechanical properties through‐thickness of FGP nanoshells can be expressed as follows[43]:

P zð Þ ¼PmþðPcPmÞ 1 2þz

h

n

#ðPcþPmÞ; ð1Þ wherePrepresents material properties such as Young's modulusE, mass densityρ, Poisson's ratioν, thermal expansion coefficientα, and moisture expansion coefficientβ. Subscriptsmandcdenote the metal- lic and ceramic constituents, respectively. The volume‐fraction of cera- mic and metal varying through‐thickness via the power‐law indexn,#

is the coefficient of the porosity distribution and letξbe the porosity coefficient. The relationship between#andξof each type of porosity distribution is expressed as follows[43]:

- Uneven porosity distribution (uneven):

#¼ξ

2 12j jz h

: ð2Þ

- Logarithmic‐uneven porosity distribution (Log‐uneven):

#¼ 12j jz h

log10 ξ 2þ1

: ð3Þ

Young’s modulus of theAl=Al2O3through‐thickness inz‐axis with two types of porosity distribution: uneven and log‐unevenðξ¼0:2) as shown inFig. 2(a), (b).

2.2. Nonlocal elastic theory

In the local plate theory (the classical plate theory), the stress ten- sor at any point is dependent on the strain tensor at that point. How- ever, the nonlocal theory assumed that the stress tensor at a point depends on the strain tensor at all points. The stress–strain relation can be written by[9]:

σijμ²r²σij¼σ^lij; ð4Þ

σ^lij¼Cijklɛ^ijαΔTβΔC

;μ¼e0l; ð5Þ

whereσ^ijis the nonlocal stress tensor,σ^lijis the local stress tensor, Cijkl is the elastic constant‐coefficient,ɛij is the local strain tensor,μ illustrates the small scale effect,lis an internal characteristic length and e0 represents a constant, ΔT and ΔCare the temperature rise and moisture concentrations through‐the‐thickness andr²¼_@^@_x²2þ_@^@_y²2

is the two‐dimensional Laplacian operator.

2.3. Four-unknown high-order shear deformation theory

In this work, the displacementfield (U1;U2;U3) in the middle sur- face of the FGP nanoshell is given as follows:

U1ðx;y;z;tÞ ¼ 1þ_R^z

1

u0ðx;y;tÞ z^@_@x^wbf zð Þ^@_@x^ws; U2ðx;y;z;tÞ ¼ 1þ_R^z

2

v⁰ðx;y;tÞ z^@_@y^wbf zð Þ^@_@y^ws; U3ðx;y;z;tÞ ¼w^bþw^s;

8>

>>

<

>>

>:

ð6Þ

withU1,U2,U3are the displacements in thex,y, andz‐axis, respec- tively;u0andv⁰are respectively the in‐plane displacements;w^bandw^s are the bending and shear components of the transverse displacement, respectively. The various four‐unknownu0,v0,w^bandw^sare functions Fig. 1.Model of the FGP nanoshell resting on EF.

Fig. 2.The Young’s modulusEwith different porosity distributions.

depending on the variablesxandy. f zð Þis a function determining the distribution of the transverse shear strains and shear stresses through‐

the‐thickness of the plate. The function f zð Þis chosen to satisfy the stress‐free BCs on the top and bottom surfaces of the shell, thus a shear correction factor is not required. In this article, we use four different forms of the functionf zð Þas follows:

- Hyperbolic sine function follows Soldatos[88](Present A):

f zð Þ ¼zhsinh z

h þzcosh 1

2 : ð7Þ

- Sinusoidal function follows Touratier[89](Present B):

f zð Þ ¼zh πsin πz

h : ð8Þ

- New sinusoidal function follows Mechab[76](Present C):

f zð Þ ¼z zcos¹₂

cos ¹₂ 1þ hsin^z_h

cos ¹₂ 1: ð9Þ

- New hyperbolic sine function follows Mechab[11](Present D):

f zð Þ ¼ 2zsinh ^z_h²2

2sinh ¹₄ þcosh ¹₄ : ð10Þ

The strain components of FGP nanoshells are deduced from the dis- placementsfield to

ɛxx¼@U1

@x þU3

R1

¼@u0

@xþw^bþw^s R1

z @²w^b

@x² @u0

R1@x

f zð Þ@²w^s

@x²; ð11Þ ɛyy¼@U2

@y þU3

R2

¼@v0

@y þw^bþw^s R2

z @²w^b

@y² @v0

R2@y

f zð Þ@²w^s

@y² ; ð12Þ ɛ^xy¼@U1

@y þ@U2

@x

¼@u0

@y þ@v0

@x2z @²w^b

@x@y @u0

2R1@y @v0

2R2@x

2f zð Þ@²w^s

@x@y; ð13Þ γxz¼@U1

@z þ@U3

@x u0

R1

¼g zð Þ@w^s

@x; ð14Þ

γyz¼@U2

@z þ@U3

@y v0

R2

¼g zð Þ@w^s

@y; ð15Þ

where Table 1

Properties of the FG material.

Material Properties

E(GPa) ρ(kg/m³) ν αK¹

βð%H2OÞ¹

Aluminumð ÞAl 70 2707 0.3 23 × 10^-6 0.44

ZirconiaðZrO2Þ 151 3000 0.3 10 × 10^-6 0

AluminaðAl2O3Þ 380 3800 0.3 7 × 10^-6 0.001

Table 2

Dimensionless deflectionw1of theAl=ZrO2doubly-curved shell resting on the EF with various power-law indexn(h¼a=20;Kw1¼100;Ks1¼10Þ.

R1 a;^Rb²

Method n¼0 n¼0:5 n¼1 n¼2 n¼5 n¼10 n¼ 1

(1,1) Present A 6.0866 7.2327 7.7936 8.2448 8.6436 8.9741 9.9810

Present B 6.0866 7.2327 7.7935 8.2448 8.6437 8.9741 9.9809

Present C 6.0866 7.2327 7.7936 8.2448 8.6436 8.9741 9.9810

Present D 6.0866 7.2327 7.7936 8.2448 8.6436 8.9741 9.9810

FSDT[90] 6.0881 7.2347 7.7952 8.2449 8.6415 8.9731 9.9848

ð5;5Þ Present A 6.1268 7.2777 7.8405 8.2930 8.6927 9.0240 10.033

Present B 6.1267 7.2776 7.8404 8.2930 8.6928 9.0240 10.033

Present C 6.1268 7.2777 7.8405 8.2930 8.6927 9.0240 10.033

Present D 6.1268 7.2777 7.8405 8.2930 8.6927 9.0240 10.033

FSDT[90] 6.1416 7.2945 7.8573 8.3088 8.7067 9.0393 10.059

ð5;1Þ Present A 5.6161 6.6855 7.2255 7.6840 8.1096 8.4384 9.3758

Present B 5.6160 6.6854 7.2254 7.6840 8.1097 8.4383 9.3757

Present C 5.6161 6.6855 7.2255 7.6840 8.1096 8.4384 9.3758

Present D 5.6161 6.6855 7.2255 7.6840 8.1096 8.4384 9.3758

FSDT[90] 5.6203 6.6903 7.2300 7.6764 8.1113 8.4412 9.3832

ð5;10Þ Present A 5.1045 6.0877 6.6004 7.0593 7.5053 7.8295 8.6923

Present B 5.1045 6.0876 6.6004 7.0593 7.5053 7.8295 8.6922

Present C 5.1045 6.0877 6.6004 7.0593 7.5053 7.8295 8.6923

Present D 5.1045 6.0877 6.6004 7.0593 7.5053 7.8295 8.6923

FSDT[90] 5.1063 6.0899 6.0899 7.0603 7.5049 7.8299 8.6964

ð5;5Þ Present A 4.5280 5.4108 5.8878 6.3387 6.7978 7.1131 7.8890

Present B 4.5280 5.4108 5.8878 6.3387 6.7979 7.1131 7.8890

Present C 4.5280 5.4108 5.8878 6.3387 6.7978 7.1131 7.8890

Present D 4.5280 5.4108 5.8878 6.3387 6.7978 7.1131 7.8890

FSDT[90] 4.5290 5.4122 5.8891 6.3392 6.7973 7.1132 7.8920

ð5;2:5Þ Present A 3.4163 4.0970 4.4900 4.8981 5.3478 5.6317 6.2320

Present B 3.4162 4.0970 4.4899 4.8981 5.3479 5.6316 6.2320

Present C 3.4163 4.0970 4.4900 4.8981 5.3478 5.6317 6.2320

Present D 3.4163 4.0970 4.4900 4.8981 5.3478 5.6317 6.2320

FSDT[90] 3.4181 4.0988 4.4918 4.8998 5.3493 5.6336 6.2359

Table 3

Dimensionless natural frequencyΩofAl=Al₂O₃doubly-curved shell resting on the EF with various power-law indexn(h¼a=10;K_w1¼500;K_s1¼50Þ.

R₁ a;^R_b²

Method n¼0 0:5 1 2 5 10 1

ð5;5Þ Present A 7.8460 7.3475 7.1049 6.9378 6.8999 6.8873 6.6426

Present B 7.8462 7.3476 7.1050 6.9377 6.8994 6.8871 6.6426

Present C 7.8460 7.3475 7.1049 6.9378 6.8999 6.8873 6.6426

Present D 7.8460 7.3475 7.1049 6.9378 6.8999 6.8873 6.6426

FSDT[90] 7.8460 7.3468 7.1054 6.9410 6.9097 6.8976 6.6423

ð5;10Þ Present A 7.7433 7.2630 7.0324 6.8802 6.8597 6.8544 6.6134

Present B 7.7435 7.2631 7.0325 6.8802 6.8592 6.8542 6.6134

Present C 7.7433 7.2630 7.0324 6.8802 6.8597 6.8544 6.6134

Present D 7.7433 7.2630 7.0324 6.8802 6.8597 6.8544 6.6134

FSDT[90] 7.7428 7.2619 7.0325 6.8831 6.8695 6.8644 6.6128

ð5;1Þ Present A 7.6616 7.1964 6.9754 6.8348 6.8270 6.8264 6.5866

Present B 7.6618 7.1965 6.9755 6.8348 6.8265 6.8262 6.5867

Present C 7.6616 7.1964 6.9754 6.8348 6.8270 6.8264 6.5866

Present D 7.6616 7.1964 6.9754 6.8348 6.8270 6.8264 6.5866

FSDT[90] 7.6598 7.1943 6.9747 6.8371 6.8363 6.8359 6.5857

ð1;1Þ Present A 7.6163 7.1652 6.9528 6.8224 6.8253 6.8272 6.5812

Present B 7.6165 7.1653 6.9529 6.8224 6.8248 6.8270 6.5813

Present C 7.6163 7.1652 6.9528 6.8224 6.8253 6.8272 6.5812

Present D 7.6163 7.1652 6.9528 6.8224 6.8253 6.8272 6.5812

FSDT[90] 7.6160 7.1640 6.9527 6.8253 6.8353 6.8374 6.5806

Table 4

Dimensionless deflectionw₂and stressσi of theAl=Al₂O₃doubly-curved nanoshell resting on the EF under sinusoidal loadða=h¼10;n¼1;ξ¼0:2;K_w¼50;K_s¼5Þ.

w2;σi

_R

1 a;^R_b²

Method Logarithmic-uneven porosity distribution Uneven porosity distribution

μ¼0 μ¼1 μ¼2 μ¼0 μ¼1 μ¼2

w2 ð5;1Þ Present A 3.2673 3.5964 4.3270 3.3662 3.6962 4.4232

Present B 3.2677 3.5968 4.3275 3.3667 3.6967 4.4237

Present C 3.2674 3.5965 4.3271 3.3663 3.6963 4.4233

Present D 3.2674 3.5965 4.3271 3.3663 3.6963 4.4233

ð5;5Þ Present A 3.0964 3.4228 4.1573 3.1932 3.5213 4.2540

Present B 3.0968 3.4232 4.1577 3.1937 3.5218 4.2545

Present C 3.0965 3.4229 4.1574 3.1933 3.5214 4.2541

Present D 3.0965 3.4229 4.1574 3.1933 3.5214 4.2541

ð5;5Þ Present A 3.3397 3.6696 4.3976 3.4410 3.7715 4.4950

Present B 3.3402 3.6700 4.3980 3.4416 3.7720 4.4955

Present C 3.3398 3.6696 4.3977 3.4412 3.7716 4.4951

Present D 3.3398 3.6696 4.3977 3.4412 3.7716 4.4951

σxx h

2 ð5;1Þ Present A 2.3811 2.6209 3.1533 2.4561 2.6969 3.2273

Present B 2.3826 2.6225 3.1552 2.4576 2.6985 3.2292

Present C 2.3813 2.6212 3.1536 2.4563 2.6971 3.2276

Present D 2.3813 2.6212 3.1536 2.4563 2.6971 3.2276

ð5;5Þ Present A 2.4155 2.6701 3.2431 2.4561 2.6969 3.2273

Present B 2.4170 2.6717 3.2449 2.4576 2.6985 3.2292

Present C 2.4158 2.6704 3.2434 2.4563 2.6971 3.2276

Present D 2.4158 2.6704 3.2434 2.4563 2.6971 3.2276

ð5;5Þ Present A 2.2589 2.4820 2.9745 2.3266 2.5500 3.0392

Present B 2.2605 2.4837 2.9764 2.3281 2.5516 3.0411

Present C 2.2592 2.4823 2.9748 2.3268 2.5502 3.0395

Present D 2.2592 2.4823 2.9748 2.3268 2.5502 3.0395

σxy^h₂

ð5;1Þ Present A 2.5065 2.7590 3.3195 2.5821 2.8353 3.3930

Present B 2.5076 2.7602 3.3208 2.5832 2.8364 3.3942

Present C 2.5067 2.7592 3.3197 2.5823 2.8355 3.3932

Present D 2.5067 2.7592 3.3197 2.5823 2.8355 3.3932

ð5;5Þ Present A 2.6589 2.9391 3.5698 2.7381 3.0194 3.6477

Present B 2.6600 2.9403 3.5711 2.7391 3.0205 3.6489

Present C 2.6591 2.9393 3.5700 2.7383 3.0196 3.6479

Present D 2.6591 2.9393 3.5700 2.7383 3.0196 3.6479

ð5;5Þ Present A 2.2487 2.4708 2.9610 2.3207 2.5436 3.0316

Present B 2.2498 2.4719 2.9623 2.3218 2.5447 3.0328

Present C 2.2489 2.4710 2.9612 2.3209 2.5437 3.0317

Present D 2.2489 2.4710 2.9612 2.3209 2.5437 3.0317

σxzð Þ0 ð5;1Þ Present A 1.0689 1.1766 1.4156 0.9706 1.0658 1.2754

Present B 1.1095 1.2213 1.4694 1.0085 1.1074 1.3251

Present C 1.0758 1.1842 1.4248 0.9772 1.0730 1.2840

Present D 1.0758 1.1842 1.4248 0.9772 1.0730 1.2840

ð5;5Þ Present A 1.0126 1.1193 1.3595 0.9213 1.0159 1.2273

Present B 1.0510 1.1618 1.4111 0.9572 1.0555 1.2751

Present C 1.0191 1.1266 1.3683 0.9274 1.0227 1.2355

Present D 1.0191 1.1266 1.3683 0.9274 1.0227 1.2355

ð5;5Þ Present A 1.0894 1.1970 1.4344 0.9884 1.0833 1.2911

Present B 1.1308 1.2424 1.4889 1.0269 1.1255 1.3414

Present C 1.0965 1.2047 1.4438 0.9950 1.0905 1.2998

Present D 1.0965 1.2047 1.4438 0.9950 1.0905 1.2998

g zð Þ ¼1@f zð Þ

@z : ð16Þ

In Eqs.(14) and (15), the transverse shear strains (γxz,γyz) are equal to zero on top (z¼ þh=2Þ, and bottom surfacesðz¼ h=2Þof the nanoshell.

From Eq.(4), the relation between nonlocal stresses and strains can be written as

σxxμ²r²σxx

σyyμ²r²σyy

σ^xyμ²r²σ^xy σ^xzμ²r²σ^xz σ^yzμ²r²σ^yz 8>

>>

>>

><

>>

>>

>>

:

9>

>>

>>

>=

>>

>>

>>

;

¼

C11 C12 0 0 0 C12 C22 0 0 0 0 0 C66 0 0 0 0 0 C55 0 0 0 0 0 C44

2 66 66 66 4

3 77 77 77 5

ɛ^xxαΔTβΔC ɛ^yyαΔTβΔC

ɛ^xy γxz

γyz

8>

>>

>>

<

>>

>>

>:

9>

>>

>>

=

>>

>>

>;

;

ð17Þ in which

C11¼C22¼ E zð Þ

1νð Þz²;C12¼ νð ÞE zz ð Þ

1νð Þz²;C66¼C55¼C44

¼ E zð Þ

2 1ð þνð ÞzÞ: ð18Þ

To obtain the governing equations of the FGP nanoshells, Hamil- ton’s principle is applied in the form as follows:

Zt₂

t₁ ðδUþδVþδWδKÞdt¼0; ð19Þ

In which:

The strain energy is expressed as δU¼Z

S

Z

þh=2

h=2

σxxδɛxxþσyyδɛyyþσxyδɛxyþσxzδγxzþσyzδγxz

dzdS;

Table 5

Dimensionless deflection w₂ and stress σi of the Al=Al₂O₃ elliptic parabolic nanoshell resting on the EF under sinusoidal load ða=h¼10;b=a¼1;μ¼ ffiffiffi

p3

;2R1¼R2¼10aÞ.

w2;σi

ðKw;KsÞ n Logarithmic-uneven porosity distribution Uneven porosity distribution

ξ¼0 ξ¼0:1 ξ¼0:4 ξ¼0 ξ¼0:1 ξ¼0:4

w2 ð100;0Þ 0 3.1528 3.1723 3.2259 3.1528 3.1988 3.3400

1 4.8339 4.8834 5.0248 4.8339 4.9525 5.3557

4 5.9182 6.0020 6.2538 5.9182 6.1225 6.9531

10 6.3226 6.4062 6.6536 6.3226 6.5254 7.3103

ð0;10Þ 0 2.4609 2.4727 2.5052 2.4609 2.4888 2.5734

1 3.3777 3.4018 3.4698 3.3777 3.4352 3.6244

4 3.8736 3.9093 4.0146 3.8736 3.9601 4.2917

10 4.0429 4.0769 4.1757 4.0429 4.1248 4.4252

ð100;10Þ 0 2.0083 2.0162 2.0377 2.0083 2.0269 2.0826

1 2.5797 2.5938 2.6331 2.5797 2.6131 2.7212

4 2.8593 2.8787 2.9354 2.8593 2.9062 3.0809

10 2.9505 2.9686 3.0206 2.9505 2.9939 3.1491

σxx h

2 ð100;0Þ 0 2.3472 2.3636 2.4088 2.3472 2.3860 2.5037

1 3.6429 3.6827 3.7964 3.6429 3.7382 4.0634

4 4.4487 4.5162 4.7213 4.4487 4.6139 5.3115

10 4.6833 4.7450 4.9317 4.6833 4.8340 5.5061

ð0;10Þ 0 1.8320 1.8424 1.8706 1.8320 1.8564 1.9290

1 2.5455 2.5654 2.6216 2.5455 2.5929 2.7499

4 2.9118 2.9415 3.0309 2.9118 2.9843 3.2784

10 2.9947 3.0197 3.0951 2.9947 3.0556 3.3330

ð100;10Þ 0 1.4951 1.5022 1.5215 1.4951 1.5118 1.5611

1 1.9441 1.9560 1.9894 1.9441 1.9724 2.0646

4 2.1494 2.1661 2.2161 2.1494 2.1901 2.3535

10 2.1855 2.1988 2.2389 2.1855 2.2179 2.3719

σxy^h₂

ð100;0Þ 0 13.8094 13.8824 14.0834 13.8094 13.9821 14.5078

1 3.9335 3.9725 4.0843 3.9335 4.0271 4.3472

4 4.7966 4.8654 5.0753 4.7966 4.9653 5.6837

10 5.0565 5.1178 5.3027 5.0565 5.2060 5.8754

ð0;10Þ 0 10.7785 10.8209 10.9368 10.7785 10.8785 11.1780

1 2.7485 2.7673 2.8203 2.7485 2.7933 2.9419

4 3.1395 3.1690 3.2581 3.1395 3.2116 3.5082

10 3.2332 3.2569 3.3279 3.2332 3.2908 3.5566

ð100;10Þ 0 8.7963 8.8231 8.8960 8.7963 8.8594 9.0462

1 2.0992 2.1100 2.1403 2.0992 2.1249 2.2088

4 2.3174 2.3336 2.3823 2.3174 2.3569 2.5184

10 2.3596 2.3715 2.4073 2.3596 2.3886 2.5310

σxzð Þ0 ð100;0Þ 0 2.5444 2.5245 2.4669 2.5444 2.4965 2.3287

1 1.7203 1.6720 1.5299 1.7203 1.6032 1.1795

4 0.9281 0.8345 0.5609 0.9281 0.7014 0.0053

10 1.1070 0.9911 0.6149 1.1070 0.8155 0.3054

ð0;10Þ 0 1.9859 1.9677 1.9157 1.9859 1.9423 1.7942

1 1.2021 1.1647 1.0564 1.2021 1.1121 0.7982

4 0.6075 0.5435 0.3601 0.6075 0.4537 0.0033

10 0.7078 0.6307 0.3859 0.7078 0.5155 0.1848

ð100;10Þ 0 1.6207 1.6044 1.5582 1.6207 1.5818 1.4520

1 0.9181 0.8881 0.8017 0.9181 0.8459 0.5993

4 0.4484 0.4002 0.2633 0.4484 0.3329 0.0023

10 0.5166 0.4593 0.2792 0.5166 0.3741 0.1315

¼ Z

S

Nxx @δu0

@x þδw^bþδw^s R1

Mxx @²δw^b

@x² 1 R1

@δu0

@x

Lxx@²δw^s

@x² þNyy @δv⁰

@y þδw^bþδw^s R2

Myy @²δw^s

@y² @δv⁰ R2@y

Lyy@²δw^s

@y² þNxy @δu0

@y þ@δv⁰

@x

Mxy 2@²δw^b

@x@y@δu0

R1@y@δv0

R2@x

2Lxy@²δw^s

@x@yþQxz@δw^s

@x þQyz@δw^s

@y

dS; ð20Þ

whereSis the area of the nanoshell.

The energy stored in the deformed elastic foundation is given by δV¼ Z

S

Rδw^bþw^s

dS; ð21Þ

whereR¼kww^bþw^s

ksr²w^bþw^s

is the reaction force of the foundation.

The work done by applied force can be obtained by δW¼Z

S

q xð ;yÞδU3þN^x₀N^TC_x @U3

@x @δU3

@x þN^y0N^TC_y @U3

@y @δU3

@y

dS ð22Þ N^TCx ¼N^TxþN^Cx;N^TCy ¼N^TyþN^Cy; ð23Þ

where q x;ð yÞ, N^x0;N^y0

, N^Tx;N^Ty

, and N^Cx;N^Cy

is the vertical forces and forces in the horizontal plane caused by mechanical, ther- mal, and moisture loads. These forces are given as follows

N^x₀¼N^y₀¼ N0; ð24Þ

N^T_x¼N^T_y¼N^T¼ Z^h₂

₂^hðC11þC12Þαð ÞzΔTdz

¼ Z ^h₂

^h₂

EðzÞ

1vð Þz αð ÞzΔTdz; ð25Þ

N^Cx¼N^Cy¼N^C¼ Z ^h₂

^h₂ðC11þC12Þβð ÞzΔCdz

¼ Z ^h₂

^h₂

E zð Þ

1vð Þz βð ÞzΔCdz: ð26Þ The kinetic energy is given by

δK¼Z

S

Z

þh=2

h=2

ρð Þz U_1δU_1þU_2δU_2þU_3δU_3

dzdS

¼Z

S

I01@u0

@t I11@²w^b

@t@xJ11@²w^s

@t@x

@δu0

@t þ I02@v0

@t I12@²w^b

@t@yJ12@²w^s

@t@y

@δv0

@t : Table 6

Comparison of dimensionless natural frequencyΩ, critical buckling loadN₀, and critical buckling temperatureΔTof theAl=Al₂O₃doubly-curved nanoshell resting on the EFða=h¼10;n¼1;ξ¼0:2;Kw¼50;Ks¼5Þ.

R1 a;^R_b²

Method Logarithmic-uneven porosity distribution Uneven porosity distribution

μ¼0 μ¼1 μ¼2 μ¼0 μ¼1 μ¼2

Ω ð5;1Þ Present A 11.8757 11.3192 10.3190 12.0697 11.5181 10.5286

Present B 11.8750 11.3185 10.3185 12.0688 11.5173 10.5281

Present C 11.8756 11.3190 10.3188 12.0695 11.5179 10.5285

Present D 11.8756 11.3190 10.3188 12.0695 11.5179 10.5285

ð5;5Þ Present A 12.1829 11.5874 10.5137 12.3754 11.7847 10.7214

Present B 12.1822 11.5868 10.5133 12.3746 11.7839 10.7209

Present C 12.1827 11.5873 10.5136 12.3752 11.7845 10.7213

Present D 12.1827 11.5873 10.5136 12.3752 11.7845 10.7213

ð5;5Þ Present A 11.7264 11.1867 10.2178 11.9182 11.3838 10.4263

Present B 11.7257 11.1860 10.2173 11.9173 11.3830 10.4258

Present C 11.7263 11.1865 10.2177 11.9180 11.3836 10.4262

Present D 11.7263 11.1865 10.2177 11.9180 11.3836 10.4262

N₀ ð5;1Þ Present A 8.9556 8.0966 6.6566 8.6733 7.8608 6.4988

Present B 8.9542 8.0954 6.6558 8.6717 7.8595 6.4979

Present C 8.9553 8.0964 6.6564 8.6730 7.8606 6.4986

Present D 8.9553 8.0964 6.6564 8.6730 7.8606 6.4986

ð5;5Þ Present A 10.0315 8.9951 7.2578 9.6628 8.6872 7.0518

Present B 10.0296 8.9935 7.2567 9.6608 8.6855 7.0506

Present C 10.0311 8.9948 7.2576 9.6624 8.6869 7.0515

Present D 10.0311 8.9948 7.2576 9.6624 8.6869 7.0515

ð5;5Þ Present A 9.2415 8.3353 6.8163 8.9498 8.0917 6.6533

Present B 9.2404 8.3344 6.8157 8.9486 8.0907 6.6527

Present C 9.2412 8.3352 6.8162 8.9495 8.0915 6.6532

Present D 9.2412 8.3352 6.8162 8.9495 8.0915 6.6532

ΔT ð5;1Þ Present A 11.6287 10.5133 8.6435 13.2981 12.0524 9.9642

Present B 11.6268 10.5117 8.6424 13.2956 12.0504 9.9628

Present C 11.6283 10.5130 8.6432 13.2976 12.0520 9.9639

Present D 11.6283 10.5130 8.6432 13.2976 12.0520 9.9639

ð5;5Þ Present A 13.0257 11.6800 9.4241 14.8153 13.3195 10.8120

Present B 13.0232 11.6779 9.4227 14.8121 13.3169 10.8102

Present C 13.0252 11.6796 9.4238 14.8147 13.3190 10.8116

Present D 13.0252 11.6796 9.4238 14.8147 13.3190 10.8116

ð5;5Þ Present A 11.9999 10.8233 8.8509 13.7220 12.4065 10.2011

Present B 11.9985 10.8221 8.8501 13.7202 12.4049 10.2000

Present C 11.9996 10.8230 8.8507 13.7216 12.4061 10.2008

Present D 11.9996 10.8230 8.8507 13.7216 12.4061 10.2008