1 Introduction

The discovery of carbon nanotubes by Iijima (1991) has inspired the promise of a new generation in diverse engineering, materials science, and reinforced composite structures due to superior mechanical and physical properties of carbon nanotubes over any other known materials. One of the most useful applications of this new material is the use of it as strong, light-weight and high-toughness fibers for nanocomposite structures. A large number of theoretical and experimental researches using carbon nanotubes as reinforcing fibers have been carried out.

Liao and Li (2001) carried molecular mechanics simulations to study the interfacial characteristics of polystyrene-nanotube interface. They noted that on relaxing the structure without applying any external force or displacement, there is a slight decrease in the nanotube diameter. Wei et al. (2001) discussed the effect of chemical bonding between the carbon nanotube and polymer on effective load transfer in the composites. They observed better load transfer in case of double site bonding and higher shear strain. The load transfer properties between carbon nanotubes and polymer using theoretical models were studied by Lau (2003). They found that the maximum shear stress for pullout of a single-walled carbon nanotube (SWCNT) is comparatively higher than that for a multi-walled carbon nanotube. Hammel et al. (2004) focused on production and improvement of vapour grown carbon fibers for composite applications. Han and Elliot (2007) presented classical molecular dynamics (MD) simulations of model polymer/carbon nanotube composites constructed by embedding an armchair SWCNT into methyl methacrylate polymer matrix. By comparing the simulation results with the macroscopic rule of mixture for composite systems, they showed that for strong interfacial interactions, there can be large deviations of the results from the rule of mixture.

There so many other researches in which the response of nanocomposite structures subjected to various loading conditions have been studied with computational techniques and simulation models (Zeng et al. 2008; Xiao et al. 2006; Esawi and Farag 2007; Lau et al. 2006; Shokrieh and Rafiee 2010; Qian et al. 2000; Haque and Ramasetty 2005; Saffar et al. 2008; Frankland et al. 2003; Villoria and Miravete 2007; Cao et al. 2010).

Continuing with the experimental investigations, Schadler et al. (Schadler et al. 1998) dispersed 5% by weight of multi-walled carbon nanotubes in an epoxy resin and cured the mixture using a hardener. The results indicated that the composite showed a marked difference between the change of properties in tension and compression due to the multi-walled carbon nanotube addition. Epoxy nanocomposites of different content of carbon nanofibers were fabricated and studied in terms of mechanical and electrical properties by Bal (2010). He found that flexural modulus and hardness increase significantly in refrigerated samples due to prevention of aggregates of nanofibers. On the basis of various testing procedures, it has been shown an improvement in stiffness of the carbon nanotube-polymer composite of up to 20–40% (Qian et al. 2000; Gong et al. 2000; Xu et al. 2002) and 200–350% (Geng et al. 2002; Cadek et al. 2002; Lozano and Barrera 2001), by addition of carbon nanotubes.

According to the above literature review, it can be concluded that carbon nanotubes are envisaged to be ideal reinforcements for composite materials with different polymers. However, Polyethylene is the simplest and the least expensive of all the polymers available. The molecular structure of polyethylene is the easiest to generate through no functional units and just one repetitive unit. At the atomic level, polyethylene is classified as amorphous polyethylene and crystalline polyethylene. In amorphous case, the structure of each monomer unit remains the same, but adjacent units are rotated around the connecting C–C bond. This type of structure leads to the same properties in all the directions.

In the current study, buckling behavior of nanocomposite beams which are reinforced by (10,10) armchair SWCNTs embedded in amorphous polyethylene is investigated based on the various elastic beam theories (Labuschange et al. 2009). Generalized differential quadrature (GDQ) method is utilized to discretize the governing differential equations along with four sets of end supported namely as simply supported-simply supported, clamped–clamped, clamped-simply supported, and clamped-free. Then the material properties calculated by the beam theories in conjunction with the rule of mixture are fitted with those obtained directly from MD simulations to extract consistent values of carbon nanotube efficiency parameters accounting for the scale-dependent material properties corresponding to both of short and long carbon nanotube reinforcements.

2 Overview of different beam theories

There are various beam theories to describe the behavior of beams. Consider a straight uniform beam with the length L and rectangular cross-section of thickness h. A coordinate system \(\left( {x,y,z} \right)\) is introduced on the central axis of the beam, whereas the x axis is taken along the length of the beam, the y axis in the width direction and the z axis is taken along the depth (height) direction. Also, the origin of the coordinate system is selected at the left end of the beam. It is assumed that the deformations of the beam take place in the x–z plane, so the displacement components \(\left( {u_{1} , u_{2} , u_{3} } \right)\) along the axis \(\left( {x,y,z} \right)\) are only dependent on the x and z coordinates and time t. In a general form, the following displacement field can be written:

$$u_{1} \left( {x,z,t} \right) = - z\frac{{\partial w\left( {x,t} \right)}}{\partial x} + \psi \left( z \right)\left( {\frac{{\partial w\left( {x,t} \right)}}{\partial x} + \varphi \left( {x,t} \right)} \right)$$
$$u_{2} \left( {x,z,t} \right) = 0$$
(1)
$$u_{3} \left( {x,z,t} \right) = w\left( {x,t} \right)$$

where w and \(\varphi\) are the transverse displacement and angular displacement of the beam, respectively, and \(\psi \left( z \right)\) is the shape function as follows:

For Euler–Bernoulli beam theory (EBT): \(\psi \left( z \right) = 0\).

For Timoshenko beam theory (TBT): \(\psi \left( z \right) = z\).

For Reddy beam theory (RBT): \(\psi \left( z \right) = z - \frac{{4z^{3} }}{{3h^{2} }}\).

3 Buckling analysis of nanocomposite beams

3.1 Constitutive equations

The stress-displacement and Euler–Lagrange relations for each type of beam theory can be expressed as

For Euler–Bernoulli beam theory:

$$\varepsilon_{xx} = \frac{{\partial u_{1} }}{\partial x} = - z\frac{{\partial^{2} w}}{{\partial x^{2} }} \to \sigma_{xx} = \frac{{ - zE_{11} }}{{1 - \nu^{2} }}\frac{{\partial^{2} w}}{{\partial x^{2} }}$$
(2-a)
$$\gamma_{xz} = \frac{{\partial u_{1} }}{\partial z} + \frac{{\partial u_{3} }}{\partial x} = 0 \to \sigma_{xz} = 0$$
(2-b)
$$\frac{{\partial^{2} M}}{{\partial x^{2} }} - P\frac{{\partial^{2} w}}{{\partial x^{2} }} = 0$$
(3)

For Timoshenko beam theory:

$$\varepsilon_{xx} = \frac{{\partial u_{1} }}{\partial x} = z\frac{\partial \varphi }{\partial x} \to \sigma_{xx} = \frac{{zE_{11} }}{{1 - \nu^{2} }}\frac{\partial \varphi }{\partial x}$$
(4-a)
$$\gamma_{xz} = \frac{{\partial u_{1} }}{\partial z} + \frac{{\partial u_{3} }}{\partial x} = \frac{\partial w}{\partial x} + \varphi \to \sigma_{xz} = G_{12} \left( {\frac{\partial w}{\partial x} + \varphi } \right)$$
(4-b)
$$\frac{\partial Q}{\partial x} - P\frac{{\partial^{2} w}}{{\partial x^{2} }} = 0$$
(5-a)
$$\frac{\partial M}{\partial x} - Q = 0$$
(5-b)

For Reddy beam theory:

$$\varepsilon_{xx} = \frac{{\partial u_{1} }}{\partial x} = z\frac{\partial \varphi }{\partial x} - \frac{{4z^{3} }}{{3h^{2} }}\left( {\frac{\partial \varphi }{\partial x} + \frac{{\partial^{2} w}}{{\partial x^{2} }}} \right) \to \sigma_{xx} = \frac{{zE_{11} }}{{1 - \nu^{2} }}\frac{\partial \varphi }{\partial x} - \frac{{4z^{3} E_{11} }}{{3h^{3} \left( {1 - \nu^{2} } \right)}}\left( {\frac{\partial \varphi }{\partial x} + \frac{{\partial^{2} w}}{{\partial x^{2} }}} \right)$$
(6-a)
$$\gamma_{xz} = \frac{{\partial u_{1} }}{\partial z} + \frac{{\partial u_{3} }}{\partial x} = \left( {1 - \frac{{4z^{2} }}{{h^{2} }}} \right)\left( {\varphi + \frac{\partial w}{\partial x}} \right) \to \sigma_{xz} = G_{12} \left( {1 - \frac{{4z^{2} }}{{h^{2} }}} \right)\left( {\varphi + \frac{\partial w}{\partial x}} \right)$$
(6-b)
$$\frac{4}{{3h^{2} }}\frac{{\partial^{2} R}}{{\partial x^{2} }} + \frac{\partial Q}{\partial x} - \frac{4}{{h^{2} }}\frac{\partial S}{\partial x} - P\frac{{\partial^{2} w}}{{\partial x^{2} }} = 0$$
(7-a)
$$\frac{\partial M}{\partial x} - \frac{4}{{3h^{2} }}\frac{\partial R}{\partial x} - Q + \frac{4}{{h^{2} }}S = 0$$
(7-b)

where

$$M = \mathop \int \nolimits z\sigma_{xx} dA$$
$$Q = \mathop \int \nolimits \sigma_{xz} dA$$
$$R = \mathop \int \nolimits z^{3} \sigma_{xx} dA$$
$$S = \mathop \int \nolimits z^{2} \sigma_{xz} dA$$

and P is the critical buckling load.

By substituting stress-displacement relations into the respective Euler–Lagrange relations, the constitutive equations corresponding to each type of beam theory can be obtained as

For Euler–Bernoulli beam theory:

$$\frac{{E_{11} I}}{{1 - \nu^{2} }}\frac{{\partial^{4} w}}{{\partial x^{4} }} - P\frac{{\partial^{2} w}}{{\partial x^{2} }} = 0$$
(8)

For Timoshenko beam theory:

$$\left( {\kappa G_{12} A - P} \right)\frac{{\partial^{2} w}}{{\partial x^{2} }} + \kappa G_{12} A\frac{\partial \varphi }{\partial x} = 0$$
(9-a)
$$- \kappa G_{12} A\frac{\partial w}{\partial x} + \frac{{E_{11} I}}{{1 - \nu^{2} }}\frac{{\partial^{2} \varphi }}{{\partial x^{2} }} - \kappa G_{12} A\varphi = 0$$
(9-b)

For Reddy beam theory:

$$\frac{{E_{11} I}}{{21\left( {1 - \nu^{2} } \right)}}\frac{{\partial^{4} w}}{{\partial x^{4} }} + \left( {\frac{{8G_{12} A}}{15} - P} \right)\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{16E_{11} I}}{{105\left( {1 - \nu^{2} } \right)}}\frac{{\partial^{3} \varphi }}{{\partial x^{3} }} + \frac{{8G_{12} A}}{15}\frac{\partial \varphi }{\partial x} = 0$$
(10-a)
$$- \frac{{16E_{11} I}}{{105\left( {1 - \nu^{2} } \right)}}\frac{{\partial^{3} w}}{{\partial x^{3} }} - \frac{{8G_{12} A}}{15}\frac{\partial w}{\partial x} + \frac{{68E_{11} I}}{{105\left( {1 - \nu^{2} } \right)}}\frac{{\partial^{2} \varphi }}{{\partial x^{2} }} - \frac{{8G_{12} A}}{15}\varphi = 0$$
(10-b)

where \(A, I\) are the cross-sectional area and moment of inertia of the beam, respectively, and \(\kappa\) is the shear correction factor.

3.2 Rule of mixture

In the present work, it is assumed that the carbon nanotube-reinforced composite is made of a mixture of (10,10) armchair SWCNT and polyethylene matrix with isotropic behavior. It has been shown that carbon nanotube-reinforced composites have anisotropic material properties (Han and Elliott 2007; Zhang and Shen 2006). On the basis of the rule of mixture, the effective values of Young’s modulus and shear modulus of carbon-nanotube-reinforced composite can be evaluated as (Shen 2009)

$$E_{11} = \vartheta_{1} V_{CNT} E_{11}^{CNT} + V_{m} E^{m}$$
(11-a)
$$\frac{{\vartheta_{2} }}{{E_{22} }} = \frac{{V_{CNT} }}{{E_{22}^{CNT} }} + \frac{{V_{m} }}{{E^{m} }}$$
(11-b)
$$\frac{{\vartheta_{3} }}{{G_{12} }} = \frac{{V_{CNT} }}{{G_{12}^{CNT} }} + \frac{{V_{m} }}{{G^{m} }}$$
(11-c)

in which \(E_{11}^{CNT} , E_{22}^{CNT} , G_{12}^{CNT}\) are longitudinal Young’s modulus, transverse Young’s modulus, and shear modulus of the carbon nanotube, respectively; \(E^{m} , G^{m}\) are Young’s modulus and shear modulus of the isotropic matrix, respectively; \(V_{CNT} , V_{m}\) are the volume fractions of carbon nanotube and matrix, respectively and are related by

$$V_{CNT} + V_{m} = 1$$
(12)

The coefficients of \(\vartheta_{1} , \vartheta_{2} , \vartheta_{3}\) are the carbon nanotube efficiency parameters to incorporate the scale-dependent characteristic of material properties which are determined with the results obtained directly from MD simulations.

4 Generalized differential quadrature method

The GDQ method is one of the most efficient numerical techniques to solve various boundary value problems. Many researchers have recently suggested the application of the generalized differential quadrature (GDQ) method to the analysis of nanostructures (Haftchenari et al. 2007; Malekzadeh and Fiouz 2007; De Rosa et al. 2008; Hu et al. 2009; Sepahi et al. 2010; Pradhan and Murmu 2010). This method has shown superb accuracy, efficiency, convenience and great potential in solving complicated partial differential equations. The basic idea of the differential quadrature method lies in the approximation of partial derivative of a function with respect to a coordinate at a discrete point as a weighted linear sum of the function values at all discrete points along that coordinate direction. Let \(\frac{{\partial^{r} f}}{{\partial x^{r} }}\) be the rth derivative of a function \(f\left( x \right)\) which can be expressed as a linear sum of the function values

$$\left. {\frac{{\partial^{r} f\left( x \right)}}{{\partial x^{r} }}} \right|_{{x = x_{P} }} = \mathop \sum \limits_{Q = 1}^{n} A_{PQ}^{\left( r \right)} f\left( {x_{P} } \right)$$
(13)

where n is the number of total discrete grid points used in the approximation process and \(A_{PQ}^{\left( r \right)}\) are weighting coefficients. The weighting coefficients of the first derivative are determined by

$$A_{PQ}^{\left( 1 \right)} = \frac{{M\left( {x_{P} } \right)}}{{\left( {x_{P} - x_{Q} } \right)M\left( {x_{Q} } \right)}} \left( { P, Q = 1,2, \ldots ,n; P \ne Q} \right)$$
(14)

where

$$M\left( {x_{P} } \right) = \mathop \prod \limits_{Q = 1;Q \ne P}^{n} \left( {x_{P} - x_{Q} } \right)$$
(15)

The weighting coefficients of higher-order derivatives can be obtained through the following recurrence relation

$$A_{PQ}^{\left( r \right)} = \left\{ {\begin{array}{l} {r\left[ {A_{PQ}^{{\left( {r - 1} \right)}} A_{PQ}^{\left( 1 \right)} - \frac{{A_{PQ}^{{\left( {r - 1} \right)}} }}{{x_{p} - x_{q} }}} \right], \quad P \ne Q} \\ { - \mathop \sum \limits_{Q = 1}^{n} A_{PQ}^{\left( r \right)}, \quad P = Q \quad \left( {P, Q = 1,2, \ldots ,n; 2 \le r \le n - 1} \right) } \\ \end{array} } \right.$$
(16)

4.1 Implementation of GDQ method into the constitutive equations

By applying the GDQ method, the discrete counterparts of constitutive differential equations corresponding to each type of beam theory at the rth given point can be expressed as

For Euler–Bernoulli beam theory:

$$\frac{{E_{11} I}}{{\left( {1 - \nu^{2} } \right)}}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 4 \right)} W_{s} - P\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} W_{s} = 0$$
(17)

For Timoshenko beam theory:

$$\left( {\kappa G_{12} A - P} \right)\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} W_{s} + \kappa G_{12} A\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} \phi_{s} = 0$$
(18-a)
$$- \kappa G_{12} A\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} W_{s} + \frac{{E_{11} I}}{{\left( {1 - \nu^{2} } \right)}}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} \phi_{s} - \kappa G_{12} A\phi_{r} = 0$$
(18-b)

For Reddy beam theory:

$$\left( {\frac{{8G_{12} A}}{15} - P} \right)\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} W_{s} + \frac{{16E_{11} I}}{{105\left( {1 - \nu^{2} } \right)}}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 3 \right)} \phi_{s} + \frac{{8G_{12} A}}{15}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} \phi_{s} = 0$$
(19-a)
$$- \frac{{16E_{11} I}}{{105\left( {1 - \nu^{2} } \right)}}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 3 \right)} W_{s} - \frac{{8G_{12} A}}{15}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} W_{s} + \frac{{16E_{11} I}}{{105\left( {1 - \nu^{2} } \right)}}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} \phi_{s} - \frac{{8G_{12} A}}{15}\phi_{r} = 0$$
(19-b)

4.2 Implementation of GDQ method into boundary conditions

Using the GDQ approximation, the discretized counterparts of different boundary conditions at the rth given point become for each type of beam theory as

For Euler–Bernoulli beam theory:

  • Simply supported-simply supported:

    $$W_{r} = 0 , \mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} W_{s} = 0\quad {\text{at edges}}\;x = 0 , L$$

  • Clamped–clamped:

    $$W_{r} = 0, \mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} W_{s} = 0\quad {\text{at edges}}\;x = 0 , L$$

  • Clamped-simply supported:

    $$W_{r} = 0 , \mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} W_{s} = 0\quad {\text{at edge}}\;x = 0$$
    $$W_{r} = 0 , \mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} W_{s} = 0\quad {\text{at edge}}\;x = L$$

  • Clamped-free:

    $$W_{r} = 0 , \mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} W_{s} = 0\quad {\text{at edge}}\;x = 0$$
    $$\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} W_{s} = 0, \mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 3 \right)} W_{s} = 0 \quad {\text{at edge}}\;x = L$$

For Timoshenko beam theory:

  • Simply supported-simply supported:

    $$W_{r} = 0, \mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} \phi_{s} = 0\quad {\text{at edges}}\;x = 0 , L$$

  • Clamped–clamped:

    $$W_{r} = 0, \phi_{r} = 0\quad {\text{at edges}}\;x = 0 , L$$

  • Clamped-simply supported:

    $$W_{r} = 0, \phi_{r} = 0\quad {\text{at edge}}\;x = 0$$
    $$W_{r} = 0, \mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} \phi_{s} = 0\quad {\text{at edges}}\;x = 0 , L$$

  • Clamped-free

    $$W_{r} = 0, \phi_{r} = 0\quad {\text{at edge }}x = 0$$
    $$\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} \phi_{s} = 0, \phi_{r} + \mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} W_{s} = 0 \quad {\text{at edge}}\;x = L$$

For Reddy beam theory:

  • Simply supported-simply supported:

    $$W_{r} = 0, \frac{68}{105}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} \phi_{s} - \frac{16}{105}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} W_{s} = 0 \quad {\text{at edges}}\;x = 0 , L$$

  • Clamped–clamped:

    $$W_{r} = 0, \phi_{r} = 0 \quad {\text{at edges}}\;x = 0 , L$$

  • Clamped-Simply supported:

    $$W_{r} = 0, \phi_{r} = 0\quad {\text{at edge}}\;x = 0$$
    $$W_{r} = 0, \frac{68}{105}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} \phi_{s} - \frac{16}{105}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} W_{s} = 0 \quad {\text{at edge}}\;x = 0$$

  • Clamped-free:

    $$W_{r} = 0, \phi_{r} = 0 \quad {\text{at edge}}\;x = 0$$
    $$\frac{68}{105}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} \phi_{s} - \frac{16}{105}\mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 2 \right)} W_{s} = 0, \phi_{r} + \mathop \sum \limits_{s = 1}^{n} A_{rs}^{\left( 1 \right)} W_{s} = 0\quad {\text{at edges}}\;x = L$$

5 Molecular dynamics simulation

The application of MD simulation considers as one of the most accurate methods to describe an atomic system which has the capability to handle simulations involving large numbers of atoms, allowing more complicated dynamic systems to be modeled in an approximately short period of time when compared with ab initio methods. Hanasaki et al. (2004) conducted a MD simulation of the molecular flow inside a modeled carbon nanotube junction as a strong gravitational field and periodic boundary conditions were applied in the flow direction. MD simulations of model polymer/carbon nanotube composites with different volume fraction were presented by Han and Elliott (2007). The simulation results supported the idea that it is possible to use carbon nanotubes to mechanically reinforce an appropriate polymer matrix, especially in the longitudinal direction of the nanotube. Bi et al. (2006) studied the thermal conductivity of SWCNTs dependent on tube length and temperature based on MD simulation. They demonstrated that the vacancy scattering on phonons is stronger than the isotropic atom which causes more reduction on lattice thermal conductivity of carbon nanotubes.

In the current study, unidirectional carbon nanotube-polymer nanocomposites are simulated using the molecular dynamics simulator “NanoHive” (Nanorex Inc. 2005). NanoHive is a free open source MD simulator which has certain features that can be used to model different loading conditions of nanostructures (Nanorex Inc. 2005). Two types of composites are considered namely as long-(10,10) SWCNT composite and short-(10,10) SWCNT composite that both of them are surrounded by amorphous polyethylene matrix (Fig. 1). As shown in Fig. 2, a simulation cell with approximate dimensions of \(5 \times 5 \times 10 \;{\text{nm}}\) is utilized for all simulations which are established using the adaptive intermolecular reactive empirical bond order (AIREBO) potential (Stuart et al. 2000). The AIREBO potential is an extension of the commonly used REBO potential developed for solid carbon and hydrocarbon molecules (Stuart et al. 2000). It includes the covalent bonding interactions represented by REBO potential together with the Lennard-Jones terms and torsional interactions as

Fig. 1
figure 1

(10,10) SWCNT embedded in an amorphous Polyethylene

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Fig. 2
figure 2

Schematic MD simulation cell for both of short-SWCNT and long-SWCNT composites

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$$U^{AIREBO} = \frac{1}{2}\mathop \sum \limits_{i} \mathop \sum \limits_{j \ne i} \left( {U_{ij}^{REBO} + U_{ij}^{LJ} + \mathop \sum \limits_{k \ne i,j} \mathop \sum \limits_{p \ne i,j,k} U_{kijp}^{Torsional} } \right)$$
(20)

The MD simulations presented here are all performed at constant temperature equal to the room temperature (300 K). The van Gunstern-Berendsen thermostat (Berendsen et al. 1984) is implemented in such a way that the scaling factor is used after each step of the MD simulation; the velocities of the atoms of system are scaled as the average kinetic energy remains approximately constant. A time step of \(0.5 fs\) is selected with about 2000 numbers of steps to simulate deformations of the MD cell under longitudinal and transverse strain.

Longitudinal and transverse strains are applied to the MD cell, respectively, by mathematically changing the coordinates of the atoms to an extended strained condition as depicted in Fig. 3. Then, using the NanoHive simulator, various time steps to relax the system of atoms to their equilibrium position are set up to enable the MD cell reaches to the equilibrium configuration. This procedure is repeated for different values of the tensile strain while for 5% value of the strain. The stress–strain curves of the MD cells are obtained which result in the values of Young’s modulus in the longitudinal and transverse directions. The values of Young’s modulus obtained directly from the MD simulations are given in Table 1 corresponding to both of longitudinal and transverse directions and different values of carbon nanotube volume fraction.

Fig. 3
figure 3

Applied longitudinal and transverse strains to the MD simulation cell

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Table 1 MD results for elastic moduli of carbon nanotube-reinforced composites
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6 Numerical results and discussion

The values of critical buckling load of (10,10) carbon nanotube-reinforced composite beams with four commonly used end supports are presented in this section corresponding to different types of beam theory and carbon nanotube volume fraction. Polyethylene is used as the matrix material with \(E^{m} = 3.22\;{\text{GPa}}\), \(\nu_{m} = 0.3\) at the room temperature. For the (10,10) armchair SWCNT as the reinforcement, it is assumed that \(E_{11}^{CNT} = 600\;{\text{GPa}}\), \(E_{22}^{CNT} = 10\;{\text{GPa}}\), \(G_{12}^{CNT} = 5\;{\text{GPa}}\), \(\nu_{CNT} = 0.19\) (Cornwell and Wille 1997; Popov et al. 2000).

Through matching the elastic moduli calculated by the rule of mixture and those of obtained directly from MD simulations, the carbon nanotube efficiency parameters are extracted which are given in Table 2 relevant to both short and long SWCNT reinforcements with various values of carbon nanotube volume fraction. It is worth mentioning that for the case of shear modulus, it is assumed that \(\vartheta_{3} = \vartheta_{2}\). With the comparison between the values of longitudinal and transverse Young’s moduli predicted by the rule of mixture and MD simulation, it is observed that with proper choosing of \(\vartheta_{1}\) and \(\vartheta_{2}\), the rule of mixture has an excellent capability to predict the elastic properties of nanocomposites.

Table 2 Proper values of carbon nanotube efficiency parameters
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The values of critical axial buckling load of composite beams reinforced by short-(10,10) SWCNT with thickness of \(h = 0.1\;{\text{m}}\) and various aspect ratios and carbon nanotube volume fractions are presented in Tables 3, 4, 5, 6 corresponding to simply supported-simply supported, clamped–clamped, clamped-simply supported, and clamped-free boundary conditions, respectively. The same results for composite beams reinforced by long-(10,10) SWCNT are tabulated in Tables 7, 8, 9, 10. It can be found from the results that the stiffness of nanoconposite beam reinforced with long-SWCNT is so higher than those reinforced with short-SWCNT.

Table 3 Critical buckling load of nanocomposite beam reinforced by short-SWCNT with simply supported-simply supported boundary conditions (\(10^{6} N\))
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Table 4 Critical buckling load of nanocomposite beam reinforced by short-SWCNT with clamped–clamped boundary conditions (\(10^{6} N\))
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Table 5 Critical buckling load of nanocomposite beam reinforced by short-SWCNT with clamped-simply supported boundary conditions (\(10^{6} N\))
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Table 6 Critical buckling load of nanocomposite beam reinforced by short-SWCNT with clamped-free boundary conditions (\(10^{6} N\))
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Table 7 Critical buckling load of nanocomposite beam reinforced by long-SWCNT with simply supported-simply supported boundary conditions (\(10^{6} N\))
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Table 8 Critical buckling load of nanocomposite beam reinforced by long-SWCNT with clamped–clamped supported boundary conditions (\(10^{6} N\))
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Table 9 Critical buckling load of nanocomposite beam reinforced by long-SWCNT with clamped-simply supported boundary conditions (\(10^{6} N\))
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Table 10 Critical buckling load of nanocomposite beam reinforced by long-SWCNT with clamped-free boundary conditions (\(10^{6} N\))
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Also it is seen that by incorporating the influence of transverse shear strains in Timoshenko and Reddy beam theories, the values of critical buckling load will be reduced from those of Euler–Bernoulli beam theory relevant to all carbon nanotube volume fraction specifically for the beams with lower aspect ratios. Furthermore, the difference between critical buckling loads predicted Timoshenko and Reddy beam theories is relatively more considerable corresponding to lower aspect ratios.

It can be observed from the results that an increase in the carbon nanotube volume fraction causes higher critical buckling load for both of short- and long-SWCNT reinforced composite beams, but it is more significant corresponding to the latter.

The extremely small dimensions of 1D nanostructures impose great challenges to many existing mechanical testing instruments, methodologies, and even theories. Calibration procedures in nanomechanical testing have been largely ignored; this may lead to different or even contrasting reports in the literature. Development of new nanomechanical testing techniques and calibration methods is greatly needed (Lin et al. 2010). With the advent of 2D nano-composite materials such as graphene, many researchs have been done in recent years (Li et al. 2005; Yang et al. 2013). For future wok, this research can be developed to 2D nanostructured reinforced composites.

7 Conclusion

In this work, buckling behavior of carbon nanotube-reinforced composite beams was investigated under four common sets of boundary conditions namely as simply supported-simply supported, clamped–clamped, clamped-simply supported, and clamped-free. Both of short- and long-SWCNT reinforcements were considered in the study based on different types of beam theory. The rule of mixture in conjunction with generalized differential quadrature method to discretize the constitutive differential equations was employed to obtain critical buckling loads of the nanocomposite beams. To select proper values of carbon nanotube efficiency parameters used in the rule of mixture, the elastic moduli relevant to both of composites with the short- and long-SWCNT reinforcements were evaluated using molecular dynamics simulation, the results of which were fitted with those obtained from the rule of mixture.

It was found that there are various carbon nanotube efficiency parameters corresponding to different values of carbon nanotube volume fraction. Moreover, it was observed that for higher values of carbon nanotube volume fraction, the stiffness of nanocomposite beam increases more in the case of long-SWCNT reinforcement compared to the short-SWCNT one.