Developments in materials engineering led to the microscopically inhomogeneous spatial composite materials named functionally graded materials (FGMs) which provide huge potential applications for various systems and devices, such as aerospace, aircraft, automobile and defense structures and the electronic devices. According to the fact that FGMs have been placed in category of composite materials, volume fractions of two or more material constituents such as a pair of ceramic–metal are supposed to change smoothly and continuously throughout the gradient directions. FGMs are fabricated to take advantage of desirable features of its constituent phases, for example, in a thermal protection system, the ceramic constituents are capable to withstand extreme temperature environments due to their better thermal resistance characteristics, while the metal constituents provide stronger mechanical performance and diminishes the possibility of catastrophic fracture. Hence, by presenting novel mechanical properties, FGMs have gained their applicability in many engineering fields, such as biomedical, nuclear, and mechanical engineering.
In addition, fast growing progress in the application of structural elements such as beams, shells and plates with micro/nano length-scale in micro (MEMS) or nano (NEMS) electro-mechanical systems, due to their outstanding chemical, mechanical, and electrical properties, led to a provocation in modelling of micro/nano scale structures. In such applications, the size-effect has a major role on dynamic response of material. After invention of carbon nanotubes by Iijima , nanoscale engineering materials have exposed to considerable attention in modern science and technology. These structures possess extraordinary mechanical, thermal, electrical and chemical performances that are superior to conventional structural materials. Therefore, the analysis of nanostructures attracts great attention by investigators based on molecular dynamics (MD) and continuum mechanics. The problem of classical theory in analysis of nanostructures is that classical continuum mechanics theory does not take care with size-effects in micro- or nano-scale structures. The classical continuum mechanics overpredicts the behaviors of micro-/nano-structures. Another way to capture size-effects is to rely on MD simulations that considered as a powerful and accurate implement to study of structural components at nanoscale. But even the MD simulation at nanoscale is computationally exorbitant for modeling nanostructures with large numbers of atoms. So, a conventional form of continuum mechanics that can capture small-scale effect is required. Eringen’s nonlocal continuum theory [2-4] is considered that includes small-scale effects with good accuracy to model micro/nano scale devices and systems. Nonlocal elasticity theory assumes that stress at a reference point is a function of strains at all neighbor points of solid. Hence, this theory could consider effects of small-scales.
For proper design of nanostructures, it is very important to take all essential characteristics of their mechanical behaviors at this submicron size. To achieve this goal, based on nonlocal constitutive equation of Eringen, many studies have been conducted attempting to develop nonlocal beam theories for predicting mechanical behavior of nanobeams. The potential of application of nonlocal Euler–Bernoulli beam theory (EBT) to materials in micro- and nano-scale has been presented by Peddieson et al.  as the first researcher to perform nonlocal theory to nano structures. Then, nonlocal elasticity theory gained considerable attention among nanotechnology society and utilization of this theory is extended in various mechanical analyses. Reddy  presented different available beam theories, containing EBT, Timoshenko’s beam theory (TBT), Reddy’s, and Levinson’s beam theories through nonlocal differential relations of Eringen. Wang and Liew  investigated static response of micro- and nano-scale structures based on nonlocal continuum mechanics using EBT and TBT. Aydogdu  employed nonlocal beam theory for bending, buckling, and vibration of nanobeams based on various beam theories. Phadikar and Pradhan  used the finite element method and Eringen’s theory to discuss bending, vibration, and buckling of beams and Kirchhoff’s plates. Civalek et al.  performed formulation of governing equations of nanobeams to discuss bending of cantilever microtubules based on differential quadrature method. Also, Civalek and Demir  proposed a nonlocal EBT to analyze the bending of microtubules. Thai  employed nonlocal higher-order beam model to discuss mechanical behavior of nanobeams. Zenkour and his colleagues [13-22] presented the nonlocal Eringen’s theory for static and dynamic responses of FG and composite nanobeams.
FG nanostructures are extensively used in MEMS/NEMS due to rapid developments in nanotechnology. Ke and Wang  employed small-scale effects on dynamic stability of FG microbeams based on TBT. Recently, Eltaher et al.  presented static and stability behaviors of FG nanobeams due to nonlocal elasticity theory. Şimşek and Yurtcu  performed analytically bending and buckling of FG nanobeam using nonlocal TBT and EBT. Ebrahimi et al. [26, 27] discussed applicability of differential transformation method in deducing of vibrational characteristics of FG size-dependent nanobeams. Also, Ebrahimi and Salari  investigated a semi-analytical method for vibrational and buckling behavior of FG nanobeams. Niknam and Aghdam  also presented natural vibration and buckling response of nonlocal FG beams resting on two-parameter elastic foundation.
Thus, a comprehensive survey in literature reveals that buckling response of FG nanobeams especially for those on elastic foundations are very limited. Various kinds of elastic foundation models for the sake of describing the interactions of the beam and foundation have proposed by scientists. Winkler or one-parameter elastic foundation is known as the simplest model which regards the foundation as a series of separated linear elastic springs without coupling effects between each other. The defect of Winkler’s parameter is the behavioral inconsistency associated to discontinuous deflections on interacted surface area of the beam. Pasternak  later introduced an incompressible vertical element as a shear layer which is physically realistic representation of elastic medium and can consider transverse shear stresses due to interaction of shear deformation of surrounding elastic medium. Thus, a more realistic and generalized representation of elastic foundation is expected through a two-parameter foundation model.
Reddy’s beam theory  relaxes the limitation on the warping of the cross sections and allows cubic variations in the longitudinal direction of the beam, so it can produce adequate accuracy when applying for beam analysis. However, a few studies have been made to present mechanical behavior of FG micro/nanobeams by using higher-order and sinusoidal (Touratier ) beam theories. Rahmani and Jandaghian  employed buckling response of FG nanobeams based on a nonlocal TOSDBT. Sahmani et al.  employed free vibration of FG nanobeams around postbuckling domain incorporating effects of surface free energy. Şimşek and Reddy  presented buckling of FG microbeam resting on two-parameter Pasternak’s foundation using modified couple stress and unified higher-order beam theories. Zhang et al.  developed a size-dependent FG beam resting on two-parameter foundation based on an improved TOSDBT and provided analytical solutions for bending, buckling and free vibration problems. Microstructure buckling analysis of FG third-order microbeams in thermal environment has been performed via modified strain gradient theory by Sahmani and Ansari . It is to be noted until now that a work analyzing buckling behavior of embedded FG nanobeams using the TOSDBT has not been yet presented.
In the current article, the non-classical TOSDBT is developed for buckling of FG nanobeam embedded on elastic foundations. Material properties of FG nanobeam will be continuously changed along the beam thickness according to two kinds of micromechanics models, namely, power-law and Mori–Tanaka models. By using the minimum potential energy principle, equilibrium equations are obtained and Navier-type solution is used to solve them. The obtained results based on TOSDBT are compared with those predicted in previously published works to verify the accuracy of the current solution. Numerical results are reported to discuss effects of gradient index, nonlocality and foundation parameters on buckling response of FG nanobeams.
MATERIALS AND METHODS
Power-law and Mori-Tanaka FGM beam models
The FG nanobeam (shown in Fig. 1) is graded according to two models. Firstly, the power-law (PL) model for FGMs can be represented as:
in which is the effective material property of the FG beam, subscripts and represent metal and ceramic, respectively, and represent volume fractions of metal and ceramic, respectively. The effective material properties such as Young’s modulus and Poisson’s ratio of FG nanobeam are expressed as:
in which represents power-law exponent that determines material distribution through the nanobeam thickness.
Secondly, Mori-Tanaka (MT) homogenization technique is also utilized to model effective material properties of the current FG nanobeam. According to MT the effective local bulk modulus and shear modulus can be represented as (Şimşek and Reddy ):
Therefore from Eq. (3), based on MT scheme, Young’s modulus and Poisson’s ratio can be represented by:
The shear modulus of FG nanobeam is defined by:
The displacement field of TOSDBT at any point of nanobeam is given by
in which , and represent longitudinal and transverse displacements, represents rotation of cross section at each point of neutral axis. Cauchy’s relations of Reddy’s beam model are given by
By using the extremum condition of the minimum potential energy principle in the form
Here denotes strain energy and denotes work done by external forces. The virtual strain energy is represented by:
Using Eq. (7) into Eq. (10) gives
in which , , represents applied axial compressive load, and are transverse and axial distributed loadings and and are linear and shear parameters of elastic foundation.
By using Eqs. (11) and (13) into Eq. (9) and setting coefficients of , and to zero, the following Euler–Lagrange equations may be represented as:
Nonlocal elasticity model for FG nanobeam
According to Eringen’s nonlocal elasticity model (Eringen and Edelen ), the stresses at each point in the solid may be represented by
in which represent components available in local stress tensor at point which are associated to strain tensor components as
In Eq. (15), the size that is related to nonlocal kernel and is Euclidean distance and is a constant in which is internal length, is a characteristic external length, and is experimentally estimated. The integral constitutive equations appeared in Eq. (16) in an equivalent differential form is represented by:
in which represents Laplacian operator. For a 1D material, the constitutive relations of nonlocal theory may be represented by:
where and represent nonlocal stress and strain, respectively. For a nonlocal FG beam, Eqs. (18) is expressed as
in which . Integrating Eqs. (19) over cross-section area, we obtain
where the cross-sectional rigidities are given by:
The nonlocal normal force is obtained by using second derivative of from Eq. (14)1 into Eq. (20) as
Eliminating from Eqs. (14)2 and (14)3 yields
Also, the nonlocal bending moment can be derived by using the above equation into Eqs. (21) and (22) as follows:
The substitution of second derivative of the nonlocal shear force from Eq. (14)3 into Eq. (23) with the aid of Eq. (24) yields in the form
RESULTS AND DISCUSSIONS
The displacement field that satisfy the simply-supported boundary conditions is represented as
in which , and denote unknown Fourier’s coefficients to be determined for each . It is to be noted that the conditions for simply-supported beam are expressed asUsing Eqs. (37) into Eqs. (34)-(36), respectively, leads to
The above equations may be summarized as:
where and is stiffness matrix. By putting this polynomial to zero, we can find buckling loads.
Here, effects of FG distribution, nonlocality effect and mode number on natural frequencies of FG nanobeam are presented. The FG nanobeam is a combination of steel and alumina (Al2O3) where their properties are reported in Table 1. The following dimensions for the beam geometry is considered: (length) = 104 nm, (width) = 103 nm (Eltaher et al. ; Rahmani and Pedram ). In addition, for better presentation of results the following dimensionless quantities are adopted (Şimşek and Reddy ):
where represents moment inertia of beam’s cross-section. For verification purpose, dimensionless buckling loads of simply-supported FG nanobeam with different nonlocal parameters and gradient indexes are compared with results reported in Eltaher et al.  and Rahmani and Jandaghian  for nonlocal EBT and nonlocal Reddy’s beam theory, respectively. In these work, the variation of Poisson’s ratio along the beam thickness is not considered and it is fixed to be 0.3. It can be observed from Table 2 that results of nonlocal Reddy’s beam theory are smaller than those of nonlocal EBT. This is attributed to the fact that Euler–Bernoulli’s beam model is unable to capture the influence of shear deformation.
The variation of dimensionless buckling loads of FG nanobeam for both PL and MT models with different gradient indexes (),
nonlocal parameters, foundation parameters and slenderness ratios are presented in Tables 3-5. The present results for MT model and PL model are referred to as MT-FGM and PL-FGM, respectively. It can be noticed from the tables that non-dimensional buckling loads predicted with respect to PL model are larger than that of MT homogenization scheme, related to the fact that for a constant gradient index, FG nanobeam becomes more flexible based on MT homogenization scheme compared to the PL model. The obtained results using MT and PL models are the same at since nanobeam is fully ceramic. From this point of view, the difference between results of these two models is significant when gradient index value is more than zero. Considering explanationsand according to Tables 3-5, it must be noted that, as gradient index increases dimensionless buckling load increases (constant nonlocal parameter) too. In addition, at a fixed gradient index the dimensionless buckling load decreases as nonlocal parameter increases. Furthermore, it should be stated that when the foundation parameters (Winkler’s and Pasternak’s parameter) increase, the non-dimensional buckling load increases which indicates the stiffening effect of foundation parameters on the FG nanobeam.
The effect of elastic foundation on non-dimensional buckling load of FG nanobeam with varying of gradient index at is presented in Fig. 2(a, b) and the variation of non-dimensional buckling load with and without elastic foundation based on both PL and MT models are compared. It is seen from the results of this figure that the dimensionless buckling loads of FG nanobeam embedded in elastic medium are larger than that of FG nanobeam without elastic foundation. This is since when the both foundation parameters increase the nanobeam becomes stiffer. Also, the MT scheme estimates lower values for the non-dimensional buckling loads compared to the power-law model. The reason is that, MT model provides smaller values for Young’s modulus than the power-law model and thus leads to a more flexible structure. Also, it is noticed from this figure that the dimensionless buckling load is prominently affected by lower values of gradient indexes. Also, increasing nonlocal parameter shows a decreasing effect on the dimensionless buckling load. So, as a general consequence, the presence of nonlocality and elastic foundation softens and stiffens the structure, respectively.
Fig. 3(a, b) shows variation of dimensionless buckling load of FG nanobeam with respect to slenderness ratio (at and )
for various values of gradient indexes used in MT model as well as PL model. It is seen that, dimensionless buckling load increases with increase in slenderness ratio. But this observation is accurate when slenderness ratio is in the range . Therefore, it can be deduced that effect of slenderness ratio on dimensionless buckling load is approximately diminishes for values greater than .
The softening effect of nonlocal parameter on the dimensionless buckling load of S-S FG nanobeams for various gradient indexes at with and without elastic foundation is shown in Fig. 4(a, b), so as nonlocal parameter growths, dimensionless buckling load reduces for all gradient indexes.
The variation of dimensionless buckling load of S-S FG nanobeam with Winkler’s parameter for different nonlocal parameters and gradient indexes is presented in Fig 5(a-d). In this figure, the MT model is adopted. It is seen that with increase of Winkler’s parameter dimensionless buckling load increases linearly for all values of gradient index. Also, it is observed that increasing gradient index yields the increment in dimensionless buckling load at constant Winkler’s and nonlocal parameters.
The variation of the dimensionless buckling load of S-S FG nanobeam with respect to Pasternak’s parameter and different gradient indexes and nonlocal parameters is presented in Fig. 6(a-d). It is observed that with increase of Pasternak’s parameter the dimensionless buckling load increases with a linear manner for all values of gradient index and nonlocal parameter. Also, it is seen that increasing gradient index results in increase of dimensionless buckling load at constant Pasternak’s parameter. Comparing this figure with Fig. 5(a-d) specifies that the influence of Pasternak’s parameter () on non-dimensional buckling load is more significant than that of Winkler’s parameter ().
In the present work, buckling analysis of size-dependent FG nanobeams embedded in two-parameter elastic foundation is performed based on nonlocal TOSDBT in conjunction with Navier analytical method. Two types of mathematical models, namely, power law and Mori-Tanaka models are considered. The nonlocal governing differential equations in elastic medium are derived by implementing the minimum potential energy principle and using nonlocal constitutive equations of Eringen. Accuracy of the results is examined using available date in the literature. The effects of small scale parameter, material graduation, foundation parameters and slenderness ratio on buckling behavior of FG nanobeams are investigated. It is observed that, with an increase in Winkler’s or Pasternak’s parameter, the beam becomes more rigid and the dimensionless buckling load of FG nanobeams increases. Also, the presence of nonlocality has a notable decreasing effect on the dimensionless buckling load of FG nanobeams, which shows the prominence of the nonlocal effect. So, it should be noted that reasonable selection of the value of the nonlocal parameter is also vital to ensure the accuracy of the nonlocal beam models. It must be pointed out that the PL and MT indexes have a remarkable effect on the buckling responses of FG nanobeam. Moreover, often the difference of the buckling loads between PL and MT models is very small, specifically at the range of lower gradient indexes. Thus, both material models reveal that with the increase of gradient index the buckling loads increase.
CONFLICT OF INTEREST
The authors declare that there is no conflict of interests regarding the publication of this manuscript.