Nonlocal strain gradient theory for bending,buckling,and vibration of viscoelastic functionally graded curved nanobeam embedded in an elastic medium

Abstract

This article investigates bending, buckling, and vibration analysis in viscoelastic functionally graded curved nanobeam embedded in an elastic medium under different boundary conditions. The stresses can be calculated based on not only the nonlocal stress field but also the strain gradient stress field according to the nonlocal strain gradient elasticity theory. The present higher order refined curved nanobeam theory which captures shear deformation influence does not need any shear correction factors. Two power-law models are used to describe the continuous variation of material properties of viscoelastic functionally graded curved nanobeam. Governing equations of nonlocal strain gradient viscoelastic functionally graded curved nanobeam are obtained using Hamilton’s principle. To establish the present model, the results are compared with those of functionally graded curved nanobeams. The effects of nonlocal parameter, length scale parameter, viscoelastic damping coefficient, spring stiffness, boundary conditions, and power-law exponent on the bending, buckling, and vibration responses of viscoelastic functionally graded curved nanobeam are discussed.

Introduction

The nanostructures have at least one nanometer in scope because they are small, as they are made of nanosized structural elements. As a result of special features of particles on the nanoscale, nanobeams, curved nanobeams, nanoshells, nanorods, and nanoplates or nanorings have many technological applications of nanotechnology and microscopic devices. Nanobeams are very strong tools with special capabilities, used as actuators, sensors, and nanoelectromechanical systems.^1–10 In controversy to the local continuum theory, based on the nonlocal elasticity theory for small-scale structures, the stress depends on all area points for a reference point. Reddy ¹¹ studied the nonlocal nonlinear formulations for bending of beams and plates using the classical and shear deformation theories. Nonlocal bending and vibration of plates using the third-order shear deformation have been discussed by Aghababaei and Reddy. ¹² Pradhan and Murmu ¹³ have studied the nanoscale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics. Murmu and Pradhan ¹⁴ have also discussed the nanoscale effect on the free in-plane vibration of nanoplates by nonlocal continuum model. Additional analytical studies on various behaviors of nanoplates^15–27 and nanobeams^6,28,29 have been presented. Qian et al. ³⁰ have discussed on the temperature independence of statistical model parameters for cleavage fracture in ferritic steels. Qian et al. ³¹ have investigated the comparison of constraint analyses with global and local approaches under uniaxial and biaxial loadings.

It is certainly worth mentioning that functionally graded materials (FGMs) have been widely used in both nano- and microstructures. The functionally graded (FG) curved beams are generally used as structural elements of light-weight and heavy load-bearing components, and they are used to satisfy the design obligations of stiffness in industrial designs.^32–38 Due to the extensive application of FG curved beams in several engineering areas, a deeper comprehension of the mechanical behavior of similar beams is required. The concept of FGM can be utilized for the management of microstructure of a material so that bending, buckling, and vibration behaviors of a nanobeam structure can be improved.^11,39–41 So, FG curved beams are the requisite components of a structure when dealing with a set of important structures in the aeronautical, mechanical, and civil industries. As a result to gorgeous engineering properties, such as high stiffness or strength-to-weight ratios, the curved beam structures probably play a serious role in the applications of nanotechnology.

Analysis of plates and beams supported on an elastic medium in most cases is related to the design and analysis of various foundations such as individual footings, combined, and mat foundations. Although methods of analysis of plates and beams on elastic foundation have been developed for a long time, up-to-date, practical application of these methods is a difficult problem. Analytical methods such as method of initial parameters and method of superposition based on Winkler’s soil model are complex and cannot be used by practicing engineers. Analysis based on the soil model with two coefficients of subgrade reactions produces results close to those obtained from the analysis based on Winkler’s foundation and never used in practical applications. Extensive studies about elastic or viscoelastic foundations can be found in the literature.^42–51

In this article, bending, buckling, and vibration behaviors of viscoelastic FG curved nanobeam embedded in an elastic medium are investigated based on nonlocal strain gradient theory. For the viscoelastic FG curved nanobeam, the material properties are graded in radial direction using two power-law forms. The nonlocal equations of motion of strain gradient viscoelastic FG curved nanobeam are derived based on Hamilton’s principle. The results are validated with those available in the literature. Effects of all parameters involving the nonlocal parameter, the structural damping of the FG curved nanobeam, coefficients of elastic foundation, and strain gradient length scale parameter on bending, buckling, and vibration behaviors of viscoelastic FG curved nanobeam are investigated separately.

Nonlocal strain gradient theory

Through theoretical and experimental observations, the nanostructures are in possession of both the stiffening and the softening behaviors. According to this theory by Aifantis,^52,53 Lim et al. ⁵⁴ offered the stress tensor ${\sigma }_{\mathrm{ij}}$ as follows

$${\sigma }_{\mathrm{ij}}={\sigma }_{\mathrm{ij}}^{(0)}-\nabla ·{\sigma }_{\mathrm{ij}}^{(1)}$$

(1)

where ${\sigma }_{\mathrm{ij}}^{(0)}$ and ${\sigma }_{\mathrm{ij}}^{(1)}$ are the nonlocal and high-order nonlocal stress tensors, respectively, and given as

$$\begin{array}{c}{\sigma }_{\mathrm{ij}}^{(0)}=C_{\mathrm{ijmn}}\underset{V}{\int }{\beta }_0(|X-X\prime |,e_{0}l){\epsilon }_{\mathrm{mn}}\mathrm{dV}\hfill \\ \multicolumn{1}{c}{{\sigma }_{\mathrm{ij}}^{(1)}={\eta }^2{C}_{\mathrm{ijmn}}\underset{V}{\int }{\beta }_1(|X-X\prime |,e_{1}l)\nabla {\epsilon }_{\mathrm{mn}}\mathrm{dV}}\end{array}$$

(2)

where $C_{\mathrm{ijmn}}$ , $\eta$ , $e_{0l}$ , and $e_{1l}$ are the elastic coefficients, strain gradient length scale parameter, and nonlocal parameters. The nonlocal functions ${\beta }_0(|X-X\prime |,e_{0}l)$ and ${\beta }_1(|X-X\prime |,e_{1}l)$ accompany the strain tensor ${\epsilon }_{\mathrm{mn}}$ and the first-order strain gradient $\nabla {\epsilon }_{\mathrm{mn}}$ , respectively. According to Eringen’s^55,56 nonlocal theory and considering $e_0=e_1=e$ , the nonlocal stresses ${\sigma }_{\mathrm{ij}}^{(0)}$ and ${\sigma }_{\mathrm{ij}}^{(1)}$ appeared in equation (2) can be expressed as ³⁴

$$\begin{array}{c}(1-\mu {\nabla }^2){\sigma }_{\mathrm{ij}}^{(0)}=C_{\mathrm{ijmn}}{\epsilon }_{\mathrm{mn}}\hfill \\ (1-\mu {\nabla }^2){\sigma }_{\mathrm{ij}}^{(1)}={\eta }^2{C}_{\mathrm{ijmn}}\nabla {\epsilon }_{\mathrm{mn}}\hfill \end{array}$$

(3)

where ${\nabla }^2=\frac{1}{R^2}\frac{{\partial }^2}{\partial {\theta }^2}$ and $\mu =(\mathrm{el}{)}^2$ . Applying the operator $(1-\mu {\nabla }^2)$ to equation (1) and then substituting equation (3), the general constitutive relation is given as

$$\begin{array}{c}(1-\mu {\nabla }^2){\sigma }_{\mathrm{ij}}=C_{\mathrm{ijmn}}(1-\zeta {\nabla }^2){\epsilon }_{\mathrm{mn}}\hfill \end{array}$$

(4)

where $\zeta ={\eta }^2$ . For the present viscoelastic FG curved nanobeam, equation (4) can be expressed as

$$(1-\frac{\mu }{R^2}\frac{{\partial }^2}{\partial {\theta }^2})\left\{\begin{array}{c}{\sigma }_{\theta \theta }\hfill \\ \multicolumn{1}{c}{{\sigma }_{\theta z}}\end{array}\right\}=(1-\frac{\zeta }{R^2}\frac{{\partial }^2}{\partial {\theta }^2})\left[\begin{array}{c}{\tilde{C}}_{\theta \theta }{ϵ}_{\theta \theta }\hfill \\ \multicolumn{1}{c}{{\tilde{C}}_{\theta z}{ϵ}_{\theta z}}\end{array}\right]$$

(5)

where the elastic coefficients ${\tilde{C}}_{\mathrm{ij}}$ for the Kelvin–Voigt viscoelastic FG curved nanobeam can be written as

$$\begin{array}{c}{\tilde{C}}_{\theta \theta }=E(z)(1+\kappa \frac{\partial }{\partial t}),{\tilde{C}}_{\theta z}=G(z)(1+\kappa \frac{\partial }{\partial t})\hfill \end{array}$$

(6)

where E is Young’s modulus, $G=\frac{E}{2(1+\nu )}$ is the shear modulus, $\nu$ is Poisson’s ratio, and $\kappa$ is the viscoelastic damping coefficient.

Basic formulation

Consider a viscoelastic FG curved nanobeam having thickness h and length L whose coordinates are illustrated in Figure 1. A refined three-variable viscoelastic FG curved nanobeam theory is used in this study. Based on this theory, the tangential displacement $u_{\theta }$ and the radial displacement $w_r$ can be written as

$$\begin{array}{c}\left\{\begin{array}{c}{u}_{\theta }\hfill \\ \multicolumn{1}{c}{w_r}\end{array}\right\}=\left[\begin{array}{c}1+\frac{z}{R}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]u+\left\{\begin{array}{c}\frac{z}{R}\frac{\partial }{\partial \theta }\hfill \\ \multicolumn{1}{c}{-1}\end{array}\right\}{w}_b+\left\{\begin{array}{c}\frac{f(z)}{R}\frac{\partial }{\partial \theta }\hfill \\ \multicolumn{1}{c}{-1}\end{array}\right\}{w}_s\hfill \end{array}$$

(7)

in which $f(z)=z-z{e}^{-2{(z/h)}^2}$ and the displacements u, $w_b$ , and $w_s$ are the functions of $(\theta ,t)$ . Also, u is the tangential midplane displacement, and $w_b$ and $w_s$ are the bending and shear components of radial displacement, respectively. Nonzero strain components associated with the displacement model (7) are obtained as follows

$$\begin{array}{c}{ϵ}_{\theta \theta }={ϵ}_{\theta }^0+z{ϵ}_{\theta }^1+f(z){ϵ}_{\theta }^2,{ϵ}_{\theta z}=g(z){\gamma }_{\theta z}^0\hfill \end{array}$$

(8)

where

$$\begin{array}{c}{ϵ}_{\theta }^0=\frac{1}{R}(-w_b-w_s+\frac{\partial u}{\partial \theta }),{ϵ}_{\theta }^1=\frac{1}{R^2}(\frac{\partial u}{\partial \theta }+\frac{{\partial }^2{w}_b}{\partial {\theta }^2})\hfill \\ {ϵ}_{\theta }^2=\frac{1}{R^2}\frac{{\partial }^2{w}_s}{\partial {\theta }^2},{\gamma }_{\theta z}^0=-\frac{1}{R}\frac{\partial w_s}{\partial \theta },g(z)=1-\frac{\mathrm{df}(z)}{\mathrm{dz}}\hfill \end{array}$$

(9)

Figure 1.

Geometry and coordinates of viscoelastic FG curved nanobeam embedded in an elastic medium.

The material properties such as Young’s modulus $E(z)$ of the viscoelastic FG curved nanobeam may be assumed according to the following two gradation models

$$\begin{array}{c}E(z)=E_1[1-(1-E_2/E_1)V_{\beta }],V_{\beta }={(\frac{z}{h}+\frac{1}{2})}^{\beta }\hfill \\ E(z)=E_1{(E_2/E_1)}^{V_{\beta }}\hfill \end{array}$$

(10)

where $\beta$ is the inhomogeneity parameter. Hamilton’s principle is employed to derive the equations of motion that can be expressed as

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}\delta (\mathcal{U}+\mathcal{F}+\mathcal{K})\mathrm{dt}=0\hfill \end{array}$$

(11)

where $\mathcal{U}$ is the strain energy, $\mathcal{F}$ is the work done by external forces, and $\mathcal{V}$ stands for the kinetic energy. They all are defined as

$$\begin{array}{c}\delta \mathcal{U}=\underset{v}{\int }({\sigma }_{\theta \theta }\delta {ϵ}_{\theta \theta }+{\sigma }_{\theta z}\delta {ϵ}_{\theta z})\mathrm{dv}\hfill \\ \multicolumn{1}{c}{\delta \mathcal{F}=\underset{0}{\overset{\alpha }{\int }}[\frac{{\mathcal{B}}^L}{R^2}\left(\frac{\partial (w_b+w_s)}{\partial \theta }\frac{\partial \delta (w_b+w_s)}{\partial \theta }\right)-(q-V_f)\delta (w_b+w_s)]\mathrm{Rd}\theta }\\ \multicolumn{1}{c}{\delta \mathcal{K}=\underset{v}{\int }\rho (z)[{\stackrel{··}{u}}_{\theta }\delta u_{\theta }+({\stackrel{··}{w}}_b+{\stackrel{··}{w}}_s)\delta (w_b+w_s)]\mathrm{dv}}\end{array}$$

(12)

where ${\mathcal{B}}^L$ is the applied buckling load and $V_f$ is the elastic foundation given by

$$V_f=K_{w}w-\frac{K_p}{R^2}\frac{{\partial }^2}{\partial {\theta }^2}w$$

(13)

where $K_w$ and $K_p$ are the linear Winkler and shear foundation stiffnesses, respectively. By incorporating equation (9) into equation (8), equations of motion are given as

$$\begin{array}{c}\frac{1}{R}\frac{\partial {\mathcal{N}}_{\theta }}{\partial \theta }+\frac{1}{R^2}\frac{\partial {\mathcal{M}}_{\theta }}{\partial \theta }=I_1\frac{{\partial }^{2}u}{\partial t^2}+I_2\frac{{\partial }^3{w}_b}{\partial \theta \partial t^2}+I_3\frac{{\partial }^3{w}_s}{\partial \theta \partial t^2}\hfill \\ -\frac{1}{R}{\mathcal{N}}_{\theta }+\frac{1}{R^2}\frac{{\partial }^2{\mathcal{M}}_{\theta }}{\partial {\theta }^2}-\frac{{\mathcal{B}}^L}{R^2}\frac{{\partial }^2(w_b+w_s)}{\partial {\theta }^2}+V_f-q\hfill \\ =I_2\frac{{\partial }^{3}u}{\partial \theta \partial t^2}+I_4\frac{{\partial }^4{w}_b}{\partial {\theta }^2\partial t^2}+I_5\frac{{\partial }^4{w}_s}{\partial {\theta }^2\partial t^2}-I_0\frac{{\partial }^2(w_b+w_s)}{\partial t^2}\hfill \\ -\frac{1}{R}{\mathcal{N}}_{\theta }+\frac{1}{R^2}\frac{{\partial }^2{\mathcal{S}}_{\theta }}{\partial {\theta }^2}-\frac{{\mathcal{B}}^L}{R^2}\frac{{\partial }^2(w_b+w_s)}{\partial {\theta }^2}+\frac{1}{R}\frac{\partial {\mathcal{Q}}_{\theta }}{\partial \theta }+V_f-q\hfill \\ =I_3\frac{{\partial }^{3}u}{\partial \theta \partial t^2}+I_5\frac{{\partial }^4{w}_b}{\partial {\theta }^2\partial t^2}+I_6\frac{{\partial }^4{w}_s}{\partial {\theta }^2\partial t^2}-I_0\frac{{\partial }^2(w_b+w_s)}{\partial t^2}\hfill \end{array}$$

(14)

where

$$\{I_1,I_2,I_3\}=\underset{-h/2}{\overset{h/2}{\int }}\rho (z)[(1+\frac{z}{R}),\frac{z}{R},\frac{f(z)}{R}](1+\frac{z}{R})\mathrm{dz}$$

(15)

$$\begin{array}{c}\{I_4,I_5\}=\underset{-h/2}{\overset{h/2}{\int }}\rho (z)[\frac{z}{R},\frac{f(z)}{R}]\frac{z}{R}\mathrm{dz},\hfill \\ \{I_0,I_6\}=\underset{-h/2}{\overset{h/2}{\int }}\rho (z)[1,\frac{f(z)}{R}]\mathrm{dz}\hfill \end{array}$$

(16)

The moments and forces explicit in equation (11) are given as

$$\begin{array}{c}\{{\mathcal{N}}_{\theta },{\mathcal{M}}_{\theta },{\mathcal{S}}_{\theta }\}=\underset{-h/2}{\overset{h/2}{\int }}\{1,z,f(z)\}{\sigma }_{\theta \theta }\mathrm{dz},\hfill \\ {\mathcal{Q}}_{\theta }=\underset{-h/2}{\overset{h/2}{\int }}g(z){\sigma }_{\theta z}\mathrm{dz}\hfill \end{array}$$

(17)

Substituting equation (8) into equation (5) and using equations (6) and (17), we get

$$\begin{array}{c}{\mathcal{N}}_{\theta }-\frac{\mu }{R^2}\frac{{\partial }^2{\mathcal{N}}_{\theta }}{\partial {\theta }^2}=(1-\frac{\zeta }{R^2}\frac{{\partial }^2}{\partial {\theta }^2})\hfill \\ [A_{11}({\epsilon }_{\theta }^0+\kappa \frac{\partial {\epsilon }_{\theta }^0}{\partial t})+B_{11}({\epsilon }_{\theta }^1+\kappa \frac{\partial {\epsilon }_{\theta }^1}{\partial t})+B_{22}({\epsilon }_{\theta }^2+\kappa \frac{\partial {\epsilon }_{\theta }^2}{\partial t})]\hfill \\ {\mathcal{M}}_{\theta }-\frac{\mu }{R^2}\frac{{\partial }^2{\mathcal{M}}_{\theta }}{\partial {\theta }^2}=(1-\frac{\zeta }{R^2}\frac{{\partial }^2}{\partial {\theta }^2})\hfill \\ [B_{11}({\epsilon }_{\theta }^0+\kappa \frac{\partial {\epsilon }_{\theta }^0}{\partial t})+A_{12}({\epsilon }_{\theta }^1+\kappa \frac{\partial {\epsilon }_{\theta }^1}{\partial t})+A_{22}({\epsilon }_{\theta }^2+\kappa \frac{\partial {\epsilon }_{\theta }^2}{\partial t})]\hfill \\ {\mathcal{S}}_{\theta }-\frac{\mu }{R^2}\frac{{\partial }^2{\mathcal{S}}_{\theta }}{\partial {\theta }^2}=(1-\frac{\zeta }{R^2}\frac{{\partial }^2}{\partial {\theta }^2})\hfill \\ [B_{22}({\epsilon }_{\theta }^0+\kappa \frac{\partial {\epsilon }_{\theta }^0}{\partial t})+A_{22}({\epsilon }_{\theta }^1+\kappa \frac{\partial {\epsilon }_{\theta }^1}{\partial t})+A_{33}({\epsilon }_{\theta }^2+\kappa \frac{\partial {\epsilon }_{\theta }^2}{\partial t})]\hfill \\ {\mathcal{Q}}_{\theta }-\frac{\mu }{R^2}\frac{{\partial }^2{\mathcal{Q}}_{\theta }}{\partial {\theta }^2}=(1-\frac{\zeta }{R^2}\frac{{\partial }^2}{\partial {\theta }^2})A_{44}({\gamma }_{\theta z}^0+\kappa \frac{\partial {\gamma }_{\theta z}^0}{\partial t})\hfill \end{array}$$

(18)

where the constants $A_{\mathrm{ij}}$ and $B_{\mathrm{ij}}$ are given as

$$\begin{array}{c}\multicolumn{1}{c}{\begin{array}{c}\{A_{11},B_{11},B_{22},A_{12},A_{22},A_{33}\}\hfill \\ =\underset{-h/2}{\overset{h/2}{\int }}E(z)[1,z,f(z),z^2,\mathrm{zf}(z),f^2(z)]\mathrm{dz}\hfill \end{array}}\\ \multicolumn{1}{c}{A_{44}=\underset{-h/2}{\overset{h/2}{\int }}G(z)g^2(z)\mathrm{dz}}\end{array}$$

(19)

Applying operator $(1-\frac{\mu }{R^2}\frac{{\partial }^2}{\partial {\theta }^2})$ to equation (14) and using equations (9) and (18), equations of motion in terms of displacements u, $w_b$ , and $w_s$ are given as

$$\begin{array}{l}\left(\frac{\partial }{\partial \theta }-\frac{\zeta }{R^2}\frac{{\partial }^3}{\partial {\theta }^3}\right)[\left(\frac{A_{11}}{R^2}+\frac{B_{11}}{R^3}\right)\left(1+\kappa \frac{\partial }{\partial t}\right)\left(-w_b-w_s+\frac{\partial u}{\partial \theta }\right)\hfill \\ +\left(\frac{B_{11}}{R^3}+\frac{A_{12}}{R^4}\right)\left(1+\kappa \frac{\partial }{\partial t}\right)\left(\frac{\partial u}{\partial \theta }+\frac{{\partial }^2{w}_b}{\partial {\theta }^2}\right)+\left(\frac{B_{22}}{R^3}+\frac{A_{22}}{R^4}\right)\left(1+\kappa \frac{\partial }{\partial t}\right)\frac{{\partial }^2{w}_s}{\partial {\theta }^2}]\hfill \\ -\left(1-\frac{\mu }{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)\left[I_1\frac{{\partial }^{2}u}{\partial t^2}+I_2\frac{{\partial }^3{w}_b}{\partial \theta \partial t^2}+I_3\frac{{\partial }^3{w}_s}{\partial \theta \partial t^2}\right]=0\hfill \\ \left(1-\frac{\zeta }{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)[\left(-\frac{A_{11}}{R}+\frac{B_{11}}{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)\left(1+\kappa \frac{\partial }{\partial t}\right)\left(-w_b-w_s+\frac{\partial u}{\partial \theta }\right)\hfill \\ +\left(-\frac{B_{11}}{R}+\frac{A_{12}}{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)\left(1+\kappa \frac{\partial }{\partial t}\right)\left(\frac{\partial u}{\partial \theta }+\frac{{\partial }^2{w}_b}{\partial {\theta }^2}\right)+\left(-\frac{B_{22}}{R}+\frac{A_{22}}{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)\left(1+\kappa \frac{\partial }{\partial t}\right)\frac{{\partial }^2{w}_s}{\partial {\theta }^2}]\hfill \\ -\left(1-\frac{\mu }{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)[\frac{{ℬ}^L}{R^2}\frac{{\partial }^2(w_b+w_s)}{\partial {\theta }^2}-V_f+q\hfill \\ +I_2\frac{{\partial }^{3}u}{\partial \theta \partial t^2}+I_4\frac{{\partial }^4{w}_b}{\partial {\theta }^2\partial t^2}+I_5\frac{{\partial }^4{w}_s}{\partial {\theta }^2\partial t^2}-I_0\frac{{\partial }^2(w_b+w_s)}{\partial t^2}]=0\hfill \\ \left(1-\frac{\zeta }{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)[\left(-\frac{A_{11}}{R}+\frac{B_{22}}{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)\left(1+\kappa \frac{\partial }{\partial t}\right)\left(-w_b-w_s+\frac{\partial u}{\partial \theta }\right)\hfill \\ +\left(-\frac{B_{11}}{R}+\frac{A_{22}}{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)\left(1+\kappa \frac{\partial }{\partial t}\right)\left(\frac{\partial u}{\partial \theta }+\frac{{\partial }^2{w}_b}{\partial {\theta }^2}\right)+\left(-\frac{B_{22}}{R}+\frac{A_{33}}{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)\hfill \\ \left(1+\kappa \frac{\partial }{\partial t}\right)\frac{{\partial }^2{w}_s}{\partial {\theta }^2}-\frac{A_{44}}{R^2}\left(1+\kappa \frac{\partial }{\partial t}\right)\frac{{\partial }^2{w}_s}{\partial {\theta }^2}]-\left(1-\frac{\mu }{R^2}\frac{{\partial }^2}{\partial {\theta }^2}\right)[\frac{{ℬ}^L}{R^2}\frac{{\partial }^2(w_b+w_s)}{\partial {\theta }^2}\hfill \\ -V_f+q+I_3\frac{{\partial }^{3}u}{\partial \theta \partial t^2}+I_5\frac{{\partial }^4{w}_b}{\partial {\theta }^2\partial t^2}+I_6\frac{{\partial }^4{w}_s}{\partial {\theta }^2\partial t^2}-I_0\frac{{\partial }^2(w_b+w_s)}{\partial t^2}]=0\hfill \end{array}$$

(20)

Solution procedure

Analytical solution for the present problem has been obtained by solving the nonlocal equations of motion of viscoelastic FG curved nanobeam with different boundary conditions:

• Simply-supported (S): $w_b=w_s={\mathcal{M}}_{\theta }={\mathcal{S}}_{\theta }=0$ at $\theta =0,\alpha$ ;

• Clamped (C): $u=w_b=w_s=0$ at $\theta =0,\alpha$ ;

• Free (F): ${\mathcal{M}}_{\theta }={\mathcal{Q}}_{\theta }=0$ at $\theta =0,\alpha$ .

The generalized displacements have been assumed as ³⁴

$$\begin{array}{c}u(\theta ,t)=\sum _{n=1}^{\infty }{U}_n\frac{d\mathcal{H}(\theta )}{d\theta }{e}^{i{\omega }_{n}t},w_b(\theta ,t)=\sum _{n=1}^{\infty }{W}_{\mathrm{bn}}\mathcal{H}(\theta )e^{i{\omega }_{n}t}\hfill \\ \multicolumn{1}{c}{w_s(\theta ,t)=\sum _{n=1}^{\infty }{W}_{\mathrm{sn}}\mathcal{H}(\theta )e^{i{\omega }_{n}t},q(\theta ,t)=\sum _{n=1}^{\infty }{q}_0\sin \left(\frac{n\pi }{\alpha }\theta \right)e^{i{\omega }_{n}t}}\end{array}$$

(21)

in which $q_0$ is the intensity load, $(U_n,W_{\mathrm{bn}},W_{\mathrm{sn}})$ are unknown Fourier coefficients, and $\mathcal{H}(\theta )$ is a function which is selected for different boundary conditions as ⁵⁷

$$\begin{array}{c}S-S:\mathcal{H}(\theta )=\sin \left(\frac{n\pi }{\alpha }\theta \right)\hfill \\ C-S:\mathcal{H}(\theta )=\sin \left(\frac{n\pi }{\alpha }\theta \right)[\cos \left(\frac{n\pi }{\alpha }\theta \right)-1]\hfill \\ C-C:\mathcal{H}(\theta )=\stackrel{2}{\sin }\left(\frac{n\pi }{\alpha }\theta \right)\hfill \\ F-F:\mathcal{H}(\theta )=\stackrel{2}{\cos }\left(\frac{n\pi }{\alpha }\theta \right)[\stackrel{2}{\sin }\left(\frac{n\pi }{\alpha }\theta \right)+1]\hfill \end{array}$$

(22)

Substituting equation (21) into equation (20) using equation (22), one can obtain

$$\begin{array}{c}\left\{\begin{array}{ccc}{\mathrm{\Gamma }}_{11}\hfill & {\mathrm{\Gamma }}_{12}\hfill & {\mathrm{\Gamma }}_{13}\hfill \\ \multicolumn{1}{c}{{\mathrm{\Gamma }}_{12}}& {\mathrm{\Gamma }}_{22}\hfill & {\mathrm{\Gamma }}_{23}\hfill \\ \multicolumn{1}{c}{{\mathrm{\Gamma }}_{13}}& {\mathrm{\Gamma }}_{23}\hfill & {\mathrm{\Gamma }}_{33}\hfill \end{array}\right\}\left\{\begin{array}{c}{U}_n\hfill \\ \multicolumn{1}{c}{W_{\mathrm{bn}}}\\ \multicolumn{1}{c}{W_{\mathrm{sn}}}\end{array}\right\}=\left\{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathcal{G}}_1}\\ \multicolumn{1}{c}{{\mathcal{G}}_2}\end{array}\right\}\hfill \end{array}$$

(23)

where the coefficients ${\mathrm{\Gamma }}_{\mathrm{ij}}$ and ${\mathcal{G}}_i$ for the viscoelastic FG curved nanobeam embedded in an elastic medium are given in Appendix 2.

For the bending problem of the viscoelastic FG curved nanobeam, the deflection and stresses are given by setting ${\mathcal{B}}^L$ to zero and ${\omega }_n=i{10}^{-2}$ . Also, to obtain the buckling loads for viscoelastic FG curved nanobeam, we set the determinant $|\mathrm{\Gamma }|$ equal to zero and $q_0=0$ , and for free vibration problem, we set ${\mathcal{B}}^L$ to zero and $q_0=0$ .

Discussions

Bending, buckling, and vibration behaviors of nonlocal strain gradient for viscoelastic FG curved nanobeam embedded in an elastic medium under various boundary conditions are studied. The viscoelastic FG curved nanobeam is composed of steel $E_1=210\mathrm{GPa}$ , ${\rho }_1=7800\mathrm{kg}/{\mathrm{m}}^3$ ; alumina $E_2=390\mathrm{GPa}$ , ${\rho }_2=3960\mathrm{kg}/{\mathrm{m}}^3$ ; and has a length $L=10\mathrm{nm}$ , $\nu =0.3$ , and $n=1$ . Figure 2 shows the variation of Young’s modulus E through the thickness of the viscoelastic FG curved nanobeam embedded in an elastic medium for different values of the power-law exponent $\beta$ according to the two gradation models: (a) model 1 and (b) model 2. For the present results, we used the dimensionless form as

$$\begin{array}{c}\overline{w}=\frac{{10}^2{E}_2{h}^3}{12q_0{L}^4}w,{\kappa }_w=\frac{L^4}{D}{K}_w,{\kappa }_p=\frac{L^2}{D}{K}_p,L=R\alpha \hfill \\ \multicolumn{1}{c}{\{{\overline{\sigma }}_{\theta \theta },{\overline{\sigma }}_{\theta z}\}=\frac{h^{2}L}{10q_0}\{{\sigma }_{\theta \theta },-{\sigma }_{\theta z}\},D=\frac{E_2{h}^3}{12(1-{\nu }^2)}}\\ \multicolumn{1}{c}{\overline{\omega }=L^2\sqrt{\frac{{\rho }_{2}h}{D}}{\omega }_n,\phi =\frac{12R^2}{h^3{E}_2}{\mathcal{B}}^L,\kappa =\overline{\kappa }{L}^2{10}^{-2}\sqrt{\frac{{\rho }_{2}h}{D}}}\end{array}$$

(24)

Figure 2.

Variation of Young’s modulus E through the thickness of the viscoelastic FG curved nanobeam embedded in an elastic medium for various values of the inhomogeneity parameter $\beta$ according to the two gradation models: (a) model 1 and (b) model 2.

All parameter values are in dimensionless form, and all figures are concerned with the boundary condition S–S, except for Figures 8, 9, and 11 which are plotted for all four boundary conditions. The bars are omitted for simplicity.

Tables 1–3 demonstrate the variation of deflection and stresses of S–S viscoelastic FG curved nanobeam embedded in an elastic medium for different length scale, opening angles, nonlocal, and Winkler–Pasternak parameters. Table 4 illustrates the vibration of S–S FG curved nanobeam embedded in an elastic medium for various opening angles, elastic medium coefficients, nonlocal, and length scale parameters. Table 5 shows the comparison with the study of Ebrahimi and Barati ³⁴ for the buckling load of S–S FG curved nanobeam for various parameters, which illustrates the agreement of our results. Tables 6 and 7 contain the values of the buckling load of S–S curved FG nanobeam embedded in an elastic medium for various elastic medium coefficients, opening angles, length scale, and nonlocal parameters.

Table 1.

Variation of nondimensional deflection of S–S viscoelastic FG curved nanobeam embedded in an elastic medium $(L/h=10,\beta =1,\kappa =10)$ .

$\zeta$	$k_w$	$k_s$	$\mu =0$				$\mu =2$
			$\alpha =\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$	$\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$
0	0	0	1.74576	1.93635	2.67364	4.58882	2.07630	2.30046	3.16269	5.36501
	10	0	1.46476	1.59661	2.06649	3.05054	1.69057	1.83626	2.34699	3.37515
	0	10	0.60336	0.62461	0.68559	0.76776	0.63849	0.65821	0.71389	0.78680
	10	10	0.56584	0.58449	0.63756	0.70802	0.59663	0.61381	0.66197	0.72419
1	0	0	1.59385	1.76874	2.44709	4.22303	1.89674	2.10276	2.89767	4.94665
	10	0	1.35629	1.48090	1.92849	2.88445	1.56959	1.70807	2.19783	3.20464
	0	10	0.58412	0.60608	0.66969	0.75679	0.62043	0.64097	0.69946	0.77716
	10	10	0.54889	0.56823	0.62379	0.69869	0.58083	0.59879	0.64953	0.71601

Table 2.

Variation of nondimensional stress ${\sigma }_{\theta \theta }$ of S–S viscoelastic FG curved nanobeam embedded in an elastic medium $(L/h=10,\beta =1,\kappa =10)$ .

$\zeta$	$k_w$	$k_s$	$\mu =0$				$\mu =2$
			$\alpha =\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$	$\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$
0	0	0	3.19225	4.62986	8.61305	15.42812	4.54266	6.57968	12.18035	21.54173
	10	0	2.67842	3.81754	6.65714	10.25625	3.69874	5.25199	9.03889	13.55199
	0	10	1.10328	1.49345	2.20862	2.5812	1.39693	1.88259	2.74941	3.15919
	10	10	1.03468	1.39753	2.05388	2.38045	1.30534	1.75561	2.54941	2.90778
1	0	0	2.91547	4.23102	7.88892	14.21522	4.15151	6.01747	11.169338	19.89033
	10	0	2.48094	3.54248	6.21708	9.70939	3.43545	4.88799	8.47172	12.88577
	0	10	1.06847	1.44981	2.15896	2.54745	1.35797	1.83427	2.69613	3.12495
	10	10	1.00402	1.35928	2.01096	2.35186	1.27129	1.71357	2.50368	2.87907

Table 3.

Variation of nondimensional stress ${\sigma }_{\theta z}$ of S–S viscoelastic FG curved nanobeam embedded in an elastic medium $(L/h=10,\beta =1,\kappa =10)$ .

$\zeta$	$k_w$	$k_s$	$\mu =0$				$\mu =2$
			$\alpha =\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$	$\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$
0	0	0	8.91277	9.34694	10.82618	13.68239	12.69399	13.29812	15.33723	19.16020
	10	0	7.47814	7.70700	8.36770	9.09573	10.33574	10.61474	11.38158	12.05376
	0	10	3.08037	3.01504	2.77612	2.28921	3.90356	3.80489	3.46199	2.80993
	10	10	2.88883	2.82138	2.58162	2.11110	3.64763	3.54824	3.21016	2.58631
1	0	0	8.13683	8.53741	9.90800	12.58999	11.59555	12.15446	14.05063	17.66318
	10	0	6.92409	7.14806	7.80827	8.59932	9.59553	9.87307	10.65712	11.44294
	0	10	2.98200	2.92545	2.71152	2.25619	3.79293	3.70496	3.39163	2.77505
	10	10	2.80214	2.74277	2.52565	2.08297	3.55084	3.46117	3.14954	2.55669

Table 4.

Nondimensional vibration of S–S FG curved nanobeam embedded in an elastic medium for various values of inhomogeneity parameter, opening angles, length scale, and nonlocal parameters $(L/h=10,k_w=k_s=10)$ .

$\beta$	$\mu$	$\zeta =0$				$\zeta =1$
		$\alpha =\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$	$\alpha =\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$
0	0	13.12771	12.56589	11.16901	9.64355	13.39184	12.80334	11.33867	9.73755
	2	12.67416	12.15878	10.87942	9.48405	12.90295	12.36405	11.02522	9.56421
0.2	0	11.84548	11.34662	10.12544	8.80721	12.06695	11.54515	10.26640	8.88477
	2	11.46589	11.00683	9.88519	8.67573	11.65725	11.17805	10.00609	8.74179
1	0	10.10794	9.69362	8.69826	7.63889	10.27709	9.84466	8.80458	7.69685
	2	9.81877	9.43572	8.51738	7.54066	9.96443	9.56558	8.60836	7.59002
3	0	9.22689	8.85917	7.97541	7.03448	9.37155	8.98823	8.06601	7.08373
	2	8.97991	8.63907	7.82136	6.95096	9.10427	8.74986	7.89883	6.99293
5	0	8.98287	8.63046	7.77925	6.86945	9.12058	8.75335	7.86554	6.91636
	2	8.74787	8.42094	7.63255	6.78988	8.86618	8.52639	7.70632	6.82987

Table 5.

Nondimensional buckling load of S–S FG curved nanobeam for various opening angles, length scale, and nonlocal parameters $(L/h=10,\beta =1)$ .

$\zeta$	$\mathrm{Theory}$	$\mu =0$			$\mu =1$			$\mu =2$
		$\alpha =\frac{\pi }{3}$	$\alpha =\frac{\pi }{2}$	$\alpha =\frac{2\pi }{3}$	$\alpha =\frac{\pi }{3}$	$\alpha =\frac{\pi }{2}$	$\alpha =\frac{2\pi }{3}$	$\alpha =\frac{\pi }{3}$	$\alpha =\frac{\pi }{2}$	$\alpha =\frac{2\pi }{3}$
0	Ebrahimi and Barati 34	5.1234	1.6111	0.4940	4.6631	1.4664	0.4497	4.2788	1.3455	0.4126
	Present study	5.1232	1.6110	0.4938	4.6629	1.4662	0.4495	4.2786	1.3453	0.4124
1	Ebrahimi and Barati 34	5.6290	1.7702	0.5428	5.1234	1.6111	0.4940	4.7011	1.4783	0.4533
	Present study	5.6288	1.7700	0.5426	5.1232	1.6109	0.4938	4.7010	1.4781	0.4531
2	Ebrahimi and Barati 34	6.1347	1.9292	0.5915	5.5836	1.7559	0.5384	5.1234	1.6111	0.4940
	Present study	6.1345	1.9290	0.5913	5.5834	1.7557	0.5382	5.1232	1.6110	0.4938
3	Ebrahimi and Barati 34	6.6403	2.0882	0.6403	6.0438	1.9006	0.5828	5.5457	1.7439	0.5347
	Present study	6.6401	2.0880	0.6401	6.0436	1.9004	0.5826	5.5455	1.7437	0.5345

Table 6.

Nondimensional buckling load of S–S FG curved nanobeam embedded in an elastic medium $(L/h=10,\beta =1)$ .

$\mu$	$k_w$	$\zeta =0$					$\zeta =1$
		$k_s$	$\alpha =\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$	$\alpha =\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$
0	0	0	10.15661	5.12103	1.61041	0.49380	11.15903	5.62645	1.76935	0.54254
	10	0	11.96162	6.13634	2.06166	0.74763	12.96403	6.64177	2.22060	0.79637
	0	10	27.97132	15.14180	6.06409	2.99899	28.97374	15.64723	6.22303	3.04773
	10	10	29.77633	16.15712	6.51534	3.25283	30.77875	16.66254	6.67428	3.30156
0.5	0	0	9.67897	4.88019	1.53468	0.47058	10.63425	5.36185	1.68614	0.51702
	10	0	11.48398	5.89551	1.98593	0.72441	12.43926	6.37717	2.13739	0.77085
	0	10	27.49369	14.90097	5.98836	2.97577	28.44896	15.38263	6.13982	3.02222
	10	10	29.29869	15.91629	6.43961	3.22960	30.25397	16.39795	6.59107	3.27605

Table 7.

Nondimensional buckling load of S–S viscoelastic FG curved nanobeam embedded in an elastic medium $(L/h=10,k_w=k_s=10,\beta =1)$ .

$\mu$	$\kappa$	$\zeta =0$				$\zeta =1$
		$\alpha =\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$	$\alpha =\pi /4$	$\pi /3$	$\pi /2$	$2\pi /3$
0	0	30.09904	16.34739	6.61055	3.31443	31.10146	16.85282	6.76949	3.36317
	0.02	29.88514	16.23955	6.57663	3.30403	30.86645	16.73433	6.73223	3.35174
	0.03	29.77819	16.18563	6.55968	3.29883	30.74895	16.67509	6.71359	3.34603
	0.04	29.67125	16.13170	6.54272	3.29363	30.63144	16.61584	6.69496	3.34032
	0.055	29.51082	16.05082	6.51728	3.28583	30.45519	16.52697	6.66702	3.33175
0.5	0	29.62140	16.10657	6.53481	3.29121	30.57668	16.58823	6.68628	3.33766
	0.02	29.41756	16.00379	6.50249	3.28130	30.35272	16.47531	6.65077	3.32677
	0.03	29.31565	15.95241	6.48633	3.27635	30.24074	16.41885	6.63302	3.32132
	0.04	29.21373	15.90102	6.47017	3.27139	30.12877	16.36239	6.61526	3.31588
	0.055	29.06085	15.82394	6.44593	3.26396	29.96079	16.27769	6.58863	3.30771

Figures 3–5 show the variations of the deflection w with the nonlocal parameter $\mu$ , strain gradient parameter (SGP) $\zeta$ , and structural damping coefficient (SDC) $\kappa$ of viscoelastic FG curved nanobeam, respectively, which indicates that, for all angles $\alpha$ , w is a monotonic increase function of Figures 3 and 5 and decrease function of Figure 4. Also, we note that for the three parameters $\mu$ , $\zeta$ , and $\kappa$ , the value of deflection w increases with the decrease in the curved angle $(\frac{\pi }{4}\le \alpha \le \frac{2\pi }{3})$ .

Figure 3.

Influence of the nonlocal parameter $\mu$ and opening angle $\alpha$ on the deflection w of viscoelastic FG curved nanobeam embedded in an elastic medium $(\zeta =2,\kappa =10,\beta =1,{\kappa }_w={\kappa }_s=10)$ .

Figure 4.

Influence of the SGP $\zeta$ and opening angle $\alpha$ on the deflection w of viscoelastic FG curved nanobeam embedded in an elastic medium $(\mu =2,\kappa =10,\beta =1,{\kappa }_w={\kappa }_s=10)$ .

Figure 5.

Influence of the SDC $\kappa$ and opening angle $\alpha$ on the deflection w of viscoelastic FG curved nanobeam embedded in an elastic medium $(\mu =\zeta =2,\beta =1,{\kappa }_w={\kappa }_s=10)$ .

The influences of the different boundary conditions and different values of the angle on the normal and shear stress components ( ${\sigma }_{\theta \theta },{\sigma }_{\theta z}$ ) through the thickness $(-\frac{1}{2}\le z\le \frac{1}{2})$ or $(-\frac{h}{2}\le \overline{z}\le \frac{h}{2})$ of viscoelastic FG curved nanobeam embedded in an elastic medium are shown in Figures 6–9, from which we can note the values of shear component ${\sigma }_{\theta z}$ . Figures 7 and 9 are symmetric about the middle surface $z=0$ at which the maximum values are attained and vanished at the curved surfaces $z=-\frac{h}{2},z=\frac{h}{2}$ of the beam for all values of $\alpha$ , and all four boundary conditions. The values of normal component ${\sigma }_{\theta \theta }$ , for different boundary conditions, are changed between $z=-\frac{h}{2}\mathrm{and}z=\frac{h}{2}$ by some way as in Figures 6 and 8 from tension at the inner surface $z=-\frac{h}{2}$ to compression at the outer surface $z=-\frac{h}{2}$ .

Figure 6.

Influence of $\alpha$ on the normal stress ${\sigma }_{\theta \theta }$ through the thickness of viscoelastic FG curved nanobeam embedded in an elastic medium $(\mu =\zeta =2,\kappa =0.5,\beta =2,{\kappa }_w={\kappa }_s=10)$ .

Figure 7.

Influence of $\alpha$ on the shear stress ${\sigma }_{\theta z}$ through the thickness of viscoelastic FG curved nanobeam embedded in an elastic medium $(\mu =\zeta =2,\kappa =0.5,\beta =2,{\kappa }_w={\kappa }_s=10)$ .

Figure 8.

Influence of different boundary conditions on the normal stress ${\sigma }_{\theta \theta }$ through the thickness of viscoelastic FG curved nanobeam embedded in an elastic medium $(\mu =\zeta =2,\kappa =10,\alpha =\frac{2\pi }{3},\beta =2,{\kappa }_w={\kappa }_s=10)$ .

Figure 9.

Influence of different boundary conditions on the shear stress ${\sigma }_{\theta z}$ through the thickness of viscoelastic FG curved nanobeam embedded in an elastic medium $(\mu =\zeta =2,\kappa =10,\alpha =\frac{2\pi }{3},\beta =2,{\kappa }_w={\kappa }_s=10)$ .

Figure 10 demonstrates the variations of the frequency of vibration $\omega$ with the SDC $\kappa$ , nonlocal and length scale parameters $\mu$ and $\zeta$ , and opening angle $\alpha$ on the vibration of viscoelastic FG curved nanobeam for two cases: the curved beam non-embedded $({\kappa }_w={\kappa }_p=0)$ and embedded $({\kappa }_w={\kappa }_p=10)$ in an elastic medium, from which the values of vibration $\omega$ decrease with the increase in $\alpha$ , and seem to be straight lines (quasi-linear).

Figure 10.

Influence of (a) SDC $\kappa$ $(\mu =\zeta =2)$ , (b) nonlocal parameter $\mu$ $(\kappa =0.5,\zeta =2)$ , and (c) SGP $\xi$ $(\mu =2,\kappa =0.5)$ and opening angle $\alpha$ on the vibration of viscoelastic FG curved nanobeam embedded and non-embedded in an elastic medium $(\beta =1)$ .

The effect is illustrated as follows: the SDC $\kappa$ , and nonlocal and length scale parameters $\mu$ and $\zeta$ , for four different boundary conditions on the variation of the vibration $\omega$ , are illustrated in Figure 11. One can note that $\omega$ is a monotonically decreasing function of $\kappa ,\mu$ , while $\zeta$ is a monotonically increasing function, and the minimum values occur at the boundary condition S–S.

Figure 11.

Influence of (a) SDC $\kappa$ $(\mu =\zeta =2)$ , (b) nonlocal parameter $\mu$ $(\zeta =2,\kappa =0.5)$ , and (c) SGP $\xi$ $(\mu =2,\kappa =0.5)$ and boundary conditions on the vibration of viscoelastic FG curved nanobeam $(\alpha =\frac{\pi }{2},\beta =1)$ .

Figure 12 shows the influence of the nonlocal and length scale parameters $\mu$ and $\zeta$ , and SDC $\kappa$ for different values of $\alpha$ on the buckling of the beam. We noted that the value of $\varphi$ decreases with $\kappa ,\mu$ , while increases with $\zeta$ ; all curves seem to be quasi-linear.

Figure 12.

Influence of the (a) nonlocal parameter $\mu$ $(\zeta =2,\kappa =10)$ , (b) SGP $\xi$ $(\mu =2,\kappa =10)$ , and (c) SDC $\kappa$ $(\mu =\zeta =2)$ and opening angle on the buckling of S–S viscoelastic FG curved nanobeam embedded in an elastic medium $(\beta =1,{\kappa }_w={\kappa }_s=10)$ .

Conclusion

In this article, the bending, buckling, and vibration responses of viscoelastic FG curved nanobeam resting on elastic foundations with different boundary conditions are investigated based on the nonlocal strain gradient theory and the Kelvin–Voigt model. The following main conclusions are drawn:

• The bending and shear components of radial displacement w are the increasing functions of the beam angle $\alpha$ , and the shear component of stress ${\sigma }_{\theta z}$ is the symmetric function about the middle surface $z=0$ , and attained its maximum values on this surface for different $\alpha$ and all boundary conditions.

• The natural frequency of vibration $\omega$ increases or decreases with the beam angle $\alpha$ for two cases: when the curved beam is embedded and non-embedded in an elastic media, moreover, the frequency $\omega$ changed slightly with the different nonlocal parameter $\mu$ , SGP $\xi$ , and SDC $\kappa$ , but for different boundary conditions, $\omega$ shows significant changes with these parameters.

• The buckling of the beam has significant changes with the beam angle $\alpha$ , for all values of different parameters $\mu$ , $\zeta$ , and $\kappa$ .