A nonlocal beam theory for bending, buckling, and vibration of nanobeams

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Abstract

A nonlocal shear deformation beam theory is proposed for bending, buckling, and vibration of nanobeams using the nonlocal differential constitutive relations of Eringen. The theory, which does not require shear correction factor, accounts for both small scale effects and quadratic variation of shear strains and consequently shear stresses through the thickness of the beam. In addition, it has strong similarities with nonlocal Euler–Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The equations of motion are derived from Hamilton’s principle. Analytical solutions of deflection, buckling load, and natural frequency are presented for a simply supported beam, and the obtained results compare well with those predicted by the nonlocal Timoshenko and Reddy beam theories.

Introduction

Since the discovery of carbon nanotubes by Iijima (1991) in 1991, nanostructures are being increasingly used due to their large Young’s modulus, yield strength, flexibility, and conductivity properties (Zhang, Wang, Duan, Xiang, & Zong, 2009). Nanostructures can be modeled using atomistic or continuum mechanics. Compared to the atomistic approach, the continuum mechanics approach is widely used due to its computational efficiency and simplicity. Due to the presence of small scale effects at the nano scale, size-dependent continuum mechanics models such as the strain gradient theory (Nix & Gao, 1998), couple stress theory (Hadjesfandiari & Dargush, 2011), modified couple stress theory (Asghari et al., 2010, Ma et al., 2008, Reddy, 2011), and nonlocal elasticity theory (Eringen, 1972, Eringen and Edelen, 1972, Eringen, 1983) are used. Among these theories, the nonlocal elasticity theory initiated by Eringen is widely used. Unlike the local theories which assume that the stress at a point is a function of strain at that point, the nonlocal elasticity theory assumes that the stress at a point is a function of strains at all points in the continuum.

Based on the nonlocal constitutive relation of Eringen, a number of paper have been published attempting to develop nonlocal beam models and apply them to analyze the bending (Aydogdu, 2009, Civalek and Demir, 2011, Reddy, 2007, Reddy and Pang, 2008, Reddy, 2010, Roque et al., 2011, Wang and Liew, 2007, Wang et al., 2008), buckling (Ansari and Sahmani, 2011, Kumar et al., 2008, Murmu and Pradhan, 2009, Senthilkumar, 2010, Senthilkumar et al., 2010, Sahmani and Ansari, 2011, Wang et al., 2006, Wang et al., 2006), and vibration (Aydogdu, 2008, Aydogdu, 2009, Benzair et al., 2008, Janghorban and Zare, 2011, Murmu and Pradhan, 2009, Wang and Varadan, 2006, Wang et al., 2007, Zhang et al., 2005) responses of nanotubes. A review on the application of nonlocal models in the modeling of carbon nanotubes and graphenes is presented by Arash and Wang (2012). All of these models were based on Euler–Bernoulli beam theory (Civalek and Demir, 2011, Kumar et al., 2008, Murmu and Pradhan, 2009, Senthilkumar, 2010, Wang and Varadan, 2006, Wang et al., 2006, Wang and Liew, 2007, Zhang et al., 2005), Timoshenko beam theory (Benzair et al., 2008, Janghorban and Zare, 2011, Murmu and Pradhan, 2009, Reddy and Pang, 2008, Reddy, 2010, Roque et al., 2011, Senthilkumar et al., 2010, Sahmani and Ansari, 2011, Wang et al., 2006, Wang et al., 2007, Wang et al., 2008), and higher-order shear deformation beam theories (Aydogdu, 2008, Aydogdu, 2009, Ansari and Sahmani, 2011, Reddy, 2007). It should be noted that the Euler–Bernoulli beam theory is suitable for slender beams. For moderately deep beams, it underestimates deflection and overestimates buckling load and natural frequency due to neglecting the shear deformation effect. The Timoshenko beam theory accounts for the shear deformation effect by assuming a constant shear strain through the thickness of the beam. Therefore, a shear correction factor is required to compensate for the difference between the actual stress state and the constant stress state. To avoid the use of shear correction factor and obtain better prediction of response of deep beam, many higher-order shear deformation theories have been developed such as the third-order shear deformation theory proposed by Reddy (2007) and the generalized beam theory proposed by Aydogdu (2009).

In this paper, a nonlocal beam theory is proposed for bending, buckling, and vibration of nanobeams. The displacement field of the proposed theory is chosen based on the following assumptions: (1) the axial and transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments; (2) the bending component of axial displacement is similar to that given by the Euler–Bernoulli beam theory; and (3) the shear component of axial displacement gives rise to the parabolic variation of shear strain and hence to shear stress through the thickness of the beam in such a way that shear stress vanishes on the top and bottom surfaces. Based on the nonlocal constitutive relations of Eringen, equations of motion of nanobeams are derived using Hamilton’s principle. Closed-form solutions of deflection, buckling load, and natural frequency are obtained for simply supported nanobeams. To illustrate the accuracy of the present theory, the obtained results are compared with those predicted by the Euler–Bernoulli beam theory, Timoshenko beam theory, and Reddy’s beam theory.

Section snippets

Kinematics

Based on the assumptions made in the preceding section, the displacement field of the present theory can be obtained asu1(x,z,t)=u(x,t)-zdwbdx+14z-53zzh2dwsdxu2(x,z,t)=0u3(x,z,t)=wb(x,t)+ws(x,t)where u is the axial displacement along the midplane of the beam; wb and ws are the bending and shear components of transverse displacement along the midplane of the beam. The nonzero strains of the proposed beam theory areεx=dudx-zd2wbdx2-fd2wsdx2,γxz=gdwsdxwheref=-z/4+5z(z/h)2/3g=1-df/dz=5/4-5(z/h)2

Constitutive relations

Analytical solution of simply supported beam

Consider a simply supported beam with length L subjected to transverse load q and axial load N0. The simply supported boundary conditions of the beam arewb=ws=Mb=Ms=0atx=0,LThe following expansions of displacements are chosen to automatically satisfy the simply supported boundary conditions in Eq. (16)wb(x,t)=n=1Wbneiωtsinαxws(x,t)=n=1Wsneiωtsinαxwhere i=-1,α=nπ/L,(Wbn,Wsn) are coefficients, and ω is the natural frequency. The transverse load q is also expanded in the Fourier sine series asq

Numerical results

In this section, analytical solutions obtained in the previous sections are presented. The obtained results are compared with those computed independently for the first time based on the Euler–Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Reddy’s beam theory (RBT) for a wide range of nonlocal parameter and thickness ratio. The closed-form solutions of the EBT, TBT, and RBT are given in the Appendix. For all calculations, the shear correction factor and Poisson’s ratio are taken

Conclusions

A nonlocal shear deformation beam theory is proposed for bending, buckling, and vibration of nanobeams. The theory accounts for a quadratic variation of the shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the beam without using shear correction factor. Based on the nonlocal differential constitutive relation of Eringen, the nonlocal equations of motion of the proposed theory are derived from Hamilton’s principle.

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