Nonlocal finite element model with eight degrees of freedom and sinusoidal shear deformation theory for bending and buckling analysis of functionally graded nanobeams
محورهای موضوعی : فصلنامه شبیه سازی و تحلیل تکنولوژی های نوین در مهندسی مکانیکMehdi Dehghan 1 , Mojtaba Esmailian 2 *
1 - Department of Mechanical Engineering, Malek-Ashtar University
2 - Faculty of Mechanical, Malek Ashtar University of Technology, Iran
کلید واژه: Nanobeam, Nonlocal finite element, Sinusoidal Shear Deformation Theory, Static analysis, DOE,
چکیده مقاله :
In the present study, an efficient nonlocal finite element model is used to investigate the buckling and bending behavior of functionally graded (FG) nanobeams. A two-node element with eight degrees of freedom is formulated according to the sinusoidal higher-order shear deformation theory. This theory assumes an accurate sinusoidal distribution of transverse shear stress in the thickness direction to provide stress-free boundary conditions without requiring shear correction factors on the top and bottom surfaces of the nanobeams. To apply size effects, the nonlocal Eringen elasticity theory is used. The material properties of FG nanobeams vary in the thickness direction as a continuous power function. Numerical results indicate the acceptable accuracy of the nonlocal finite element model. Additionally, the effects of various parameters, such as FG material power law index, length-to-thickness aspect ratio, and nonlocal parameter, on the critical buckling load and deflection of FG nanobeams are investigated.
In the present study, an efficient nonlocal finite element model is used to investigate the buckling and bending behavior of functionally graded (FG) nanobeams. A two-node element with eight degrees of freedom is formulated according to the sinusoidal higher-order shear deformation theory. This theory assumes an accurate sinusoidal distribution of transverse shear stress in the thickness direction to provide stress-free boundary conditions without requiring shear correction factors on the top and bottom surfaces of the nanobeams. To apply size effects, the nonlocal Eringen elasticity theory is used. The material properties of FG nanobeams vary in the thickness direction as a continuous power function. Numerical results indicate the acceptable accuracy of the nonlocal finite element model. Additionally, the effects of various parameters, such as FG material power law index, length-to-thickness aspect ratio, and nonlocal parameter, on the critical buckling load and deflection of FG nanobeams are investigated.
[1] Eringen, A. C., 1972, "Nonlocal polar elastic continua," International journal of engineering science, 10 (1), pp. 1-16.
[2] Eringen, A. C., 1983, "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves," Journal of applied physics, 54 (9), pp. 4703-4710.
[3] Mindlin, R. D., Microstructure in linear elasticity. Columbia University New York, 1963.
[4] Papargyri-Beskou, S., Tsepoura, K., Polyzos, D., and Beskos, D., 2003, "Bending and stability analysis of gradient elastic beams," International Journal of solids and structures, 40 (2), pp. 385-400.
[5] Yang, F., Chong, A., Lam, D. C. C., and Tong, P., 2002, "Couple stress based strain gradient theory for elasticity," International journal of solids and structures, 39 (10), pp. 2731-2743.
[6] Askes, H. and Aifantis, E. C., 2009, "Gradient elasticity and flexural wave dispersion in carbon nanotubes," Physical Review B, 80 (19), p. 195412.
[7] Ghayesh, M. H. and Farajpour, A., 2019, "A review on the mechanics of functionally graded nanoscale and microscale structures," International Journal of Engineering Science, 137 pp. 8-36.
[8] Thai, H.-T. and Vo, T. P., 2012, "A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams," International Journal of Engineering Science, 54 pp. 58-66.
[9] Thai, H.-T., 2012, "A nonlocal beam theory for bending, buckling, and vibration of nanobeams," International Journal of Engineering Science, 52 pp. 56-64.
[10] Pham, Q.-H., Malekzadeh, P., Tran, V. K., and Nguyen-Thoi, T., 2023/04/01 2023, "Free vibration analysis of functionally graded porous curved nanobeams on elastic foundation in hygro-thermo-magnetic environment," Frontiers of Structural and Civil Engineering, 17 (4), pp. 584-605.
[11] Pham, Q.-H. and Nguyen, P.-C., 2022, "Effects of size-dependence on static and free vibration of FGP nanobeams using finite element method based on nonlocal strain gradient theory," Steel and Composite Structures, An International Journal, 45 (3), pp. 331-348.
[12] Uzun, B., Civalek, Ö., and Yaylı, M. Ö., 2023, "Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions," Mechanics Based Design of Structures and Machines, 51 (1), pp. 481-500.
[13] Chaht, F. L., Kaci, A., Houari, M. S. A., Tounsi, A., Bég, O. A., and Mahmoud, S., 2015, "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect," Steel and Composite Structures, 18 (2), pp. 425-442.
[14] Torkan, E., Pirmoradian, M., & Hashemian, M. (2019). Dynamic instability analysis of moderately thick rectangular plates influenced by an orbiting mass based on the first-order shear deformation theory. Modares Mechanical Engineering, 19(9), 2203-2213.
[15] Torkan, E., Pirmoradian, M., & Hashemian, M. (2017). Occurrence of parametric resonance in vibrations of rectangular plates resting on elastic foundation under passage of continuous series of moving masses. Modares Mechanical Engineering, 17(9), 225-236.
[16] Pirmoradian, M., & Karimpour, H. (2017). Nonlinear effects on parametric resonance of a beam subjected to periodic mass transition. Modares Mechanical Engineering, 17(1), 284-292.
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Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering 17 (1) (2025) 0005~0014 DOI 10.71939/jsme.2025.1204391
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Research article
Nonlocal finite element model with eight degrees of freedom and sinusoidal shear deformation theory for bending and buckling analysis of functionally graded nanobeams
Mehdi Dehghan, Mojtaba Esmaielian*,
Faculty of Mechanics, Malek Ashtar University of Technology, Shahin Shahr, Iran
* Mojtaba@mut-es.ac.ir
(Manuscript Received --- 19 Apr. 2025; Revised --- 03 June 2025; Accepted --- 23 June 2025)
Abstract
In the present study, an efficient nonlocal finite element model is used to investigate the buckling and bending behavior of functionally graded (FG) nanobeams. A two-node element with eight degrees of freedom is formulated according to the sinusoidal higher-order shear deformation theory. This theory assumes an accurate sinusoidal distribution of transverse shear stress in the thickness direction to provide stress-free boundary conditions without requiring shear correction factors on the top and bottom surfaces of the nanobeams. To apply size effects, the nonlocal Eringen elasticity theory is used. The material properties of FG nanobeams vary in the thickness direction as a continuous power function. Numerical results indicate the acceptable accuracy of the nonlocal finite element model. Additionally, the effects of various parameters, such as FG material power law index, length-to-thickness aspect ratio, and nonlocal parameter, on the critical buckling load and deflection of FG nanobeams are investigated.
Keywords: FG nanobeam, Nonlocal finite element, Sinusoidal Shear Deformation Theory, Static analysis.
1- Introduction
Nowadays, nanostructures such as nanorods, nanobeams, and nanoshells have diverse applications due to their outstanding electrical, chemical, thermal, and mechanical properties.
Micro/nano electromechanical systems (MEMS/NEMS) and nanoactuators are among these applications in which size effects are very important. Therefore, considering size effects in analyzing the mechanical behavior of these nanostructures is essential for better understanding their behavior and achieving appropriate designs.
It should be noted that since the classical continuum mechanics theory ignores size effects, it is not suitable for nanostructures. To overcome this issue, non-classical continuum theories, such as nonlocal elasticity theory [1, 2], strain gradient theory (SGT) [3, 4], modified coupled stress theory (MCST) [5], and nonlocal strain gradient theory (NSGT) [6], have been developed based on material size-dependent parameters.
However, investigating the effects of size dependence on the mechanical behavior of functionally graded (FG) materials with micro/nano structures should include the internal and external dimensions, which are always of fundamental importance. Unlike the classical elasticity theory, in nonlocal elasticity theory, stress at the reference point depends not only on the strain at the reference point but also on the strain at the points of the entire domain [2].
Many studies have been conducted so far based on the Eringen nonlocal elasticity theory to accurately predict the static behavior, free vibration, and buckling of homogeneous FG nanobeams. Ghayesh and Farajpour [7] reviewed studies on micro- and nano-scale structures made of FG materials. Thai and Vo [8] investigated the vibrations, buckling, and bending of nanobeams using an analytical solution based on the nonlocal sinusoidal shear deformation theory (SSDT). Thai [9] proposed a new nonlocal third-order shear deformation theory (TSDT) for the analysis of vibrations, buckling and bending of simply supported nanobeams using Eringen’s nonlocal differential constitutive relations.
Analytical solutions are generally limited to simple geometries, specific properties of FG material, boundary conditions, and loadings. Therefore, numerical methods such as finite element, isogeometric analysis (IGA), and meshless method (MM) have been used to analyze the complex behavior of size-dependent FG nanostructures. The capacity and effectiveness of these methods have been investigated in a wide range of complex applications [10-12].
In existing literature, studies on nanobeams utilizing the finite element method (FEM) remain limited, despite the method's advantages in utilizing complex geometry, loading, and boundary conditions as well as arbitrary grading properties. To the best of the authors’ knowledge, no publication currently provides a detailed examination of the static bending and buckling responses of functionally graded (FG) nanobeams with arbitrary FG material distributions using a finite element model based on sinusoidal higher-order nonlocal beam theories. Consequently, the primary objective and novelty of this paper is to propose an efficient finite element model to explore the bending and buckling behavior of FG nanobeams.
In the present study, an efficient finite element model is used to investigate the buckling and bending behavior of an FG nanobeam. The analysis is performed using a two-node beam element (with four degrees of freedom at each node) and sinusoidal shear deformation theory. The proposed model provides an accurate sinusoidal distribution of shear stress at each section of the beam without requiring a shear correction factor. The material properties of the nanobeam are considered to be functionally graded along the thickness direction and their variations are assumed to follow power law.
2- Governing equations
2-1- Material properties
A nanobeam with thickness 
, length 
, and width 
made of two distinct materials (metal and ceramic) is considered. The coordinate system for the FG nanobeam is shown in Fig. 1.
|
Fig. 1 Geometry and coordinate system of the FG nanobeam |
The material properties of the FG nanobeam vary along the thickness direction according to a power function as follows (Fig. 2):
| (1) |
Where 
and 
are the material properties corresponding respectively to the bottom and top surfaces of the FG nanobeam; 
is the material distribution parameter, which is greater than or equal to zero. For the sake of simplicity, the Poisson's ratio of the beam is assumed to be constant in this study.
2-2- Nonlocal elasticity theory
Nonlocal theories are based on size-dependent continuum mechanics that accounts for small-scale effects in the constitutive equations. Unlike the classical elasticity theory, in nonlocal elasticity theory, stress at a reference point depends not only on the strain at that point but also on the strain at all points of the body [2].
|
Fig. 2 Functionally graded distribution of the material properties along the thickness |
The nonlocal stress tensor, 
, at a point can be written as:
| (2) |
where 
is the Laplacian operator in the two-dimensional Cartesian coordinate system, 
is the classical stress tensor at some point related to the strain tensor by Hooke's law, and 
is the nonlocal parameter that includes small-scale effects (
is the characteristic constant for each material and 
is the internal characteristic length). The value of the nonlocal parameter (
) is important when the nonlocal elasticity theory is used.
For an isotropic FG nanobeam, the nonlocal constitutive relation in Eq. (2) can be rewritten as follows [13]:
| (3) |
where 
and 
are the axial stress and the transverse shear stress, and 
is the stiffness coefficient(s) that is correlated with geometric constants. Moreover, by setting 
, the constitutive relation for the classical (local) theory is obtained.
| (4) |
2-3- Sinusoidal Beam Theory Based on Nonlocal Elasticity
In this study, a quasi-two-dimensional sinusoidal shear deformation theory is considered for FG nanobeams considering transverse shear deformation. The displacement field in this theory can be derived as follows:
| (5) |
where 
and 
are the axial and transverse displacement of the neutral axis of the beam web, and 
is the rotation of the cross section perpendicular to the neutral axis as a result of transverse shear deformation. The following points can be derived from the above equation:
- The axial displacement consists of extension, bending and shear components;
- The bending component of axial displacement is similar to that given by the Euler–Bernoulli beam theory;
A sinusoidal shear deformation function that does not require a shear correction factor is used as follows [8]:
| (6) |
This equation satisfies the condition that the shear stress on the top and bottom surfaces of the beam is negligible [8]. The non-zero strains in the beam theory can be expressed as:
| (7) |
By rewriting the short form of the strain components, one obtains
| (8) |
where
| (9) |
Now, using the principle of minimum total potential energy, the governing equations can be written as follows [14-16]:
| (10) |
where 
is the total potential energy, 
is the partial change in strain energy, and 
is the variation of the work done by external forces. The change in strain energy is defined in terms of stress resultants as follows:
|
(11) |
Moreover, 
and 
are respectively the axial force, bending moment, shear moment, and shear force, which are given by the following equations:
| (12) |
The first variation of the work done by the compressive force is equal to
| (13) |
where 
and 
are the transverse and axial loads, respectively. By substituting Eqs. (11)-(13) into Eq. (10) and integrating by parts and collecting the coefficients of 
, 
and 
, the equations of motion for the sinusoidal nanobeam can be written as follows:
| (14) |
where the cross-sectional coefficients are expressed as follows:
(15) |
Using the total potential energy variation, the weak form of the governing equations is obtained as:
(16) |
3- Finite element formulation
Recently, an efficient nonlocal finite element model has been used to study the buckling and bending behavior of FG nanobeams based on the modified high-order shear deformation theory. As can be seen in Fig. 3, the element has two nodes and eight degrees of freedom. The vector components of nodal displacement are given as follows:
| (17) |
The unknown components 
and 
are approximated using the linear Lagrange interpolation function with continuity 
. However, the cubic Hermite interpolation functions with continuity order 
are used to approximate the component 
.
The generalized displacements in each element can be written as:
| (18) |
|
Fig. 3 Two-node element of the nanobeam with corresponding DOFs |
The classical interpolation functions are given as:
| (19) |
where
| (20) |
In the above functions, 
is the local coordinate of the element, as shown in Fig. 3. By inserting (18) into the generalized strain vectors of (7), one obtains:
| (21) |
where 
is the degree-of-freedom vector of the nanobeam element, as defined in (17); 
is an 8 × 1 matrix, representing the shape functions 
and their derivatives, such that
(22) |
Using the strain-displacement relations of the above equation, one can rewrite (16) as follows:
(23) |
The final governing equation system is expressed in the matrix form for static and buckling analyses of the nanobeam as:
Static analysis: assuming the application of a transverse load, , on the beam top surface, the following equation is obtained:
| (24) |
Buckling analysis: an axial load, , is applied to the beam centerline and the following equation is obtained:
| (25) |
where 
is the reference stiffness matrix, 
is the reference geometric stiffness matrix, 
is the force vector, and 
is the degrees-of-freedom vector in the reference system of the FG nanobeam. They can be obtained through the assembly of element-related matrices, which are defined as follows
| |
| (26) |
Simply supported (SS) boundary conditions are considered at the two ends of the nanobeam as presented in Table 1.
