Determination of the Appropriate Gradient Elasticity Theory for Bending Analysis of Nano-beams by Considering Boundary Conditions Effect

Abstract

In the present paper, a critique study on some models available in the literature for bending analysis of nano-beams using the gradient elasticity theory is accomplished. In nonlocal elasticity models of nano-beams, the size effect has not been properly considered in governing equations and boundary conditions. It means that in these models, because of replacing of the size effect with the inertia gradient effect, the size dependency has been ignored in bending analysis of nano-beams. Therefore, as the beam dimensions increase in comparison to its material length scale parameter, the obtained solution based on the gradient elasticity theory (either in the nonlocal elasticity theory or the strain gradient elasticity theory) should converge to the classical elasticity solution. Hence, satisfying of boundary conditions is a crucial point. In this paper, governing equations and boundary conditions are presented based on two gradient elasticity theories (i.e., nonlocal elasticity and strain gradient elasticity theories). Also, boundary conditions in strain gradient elasticity theory are modified based on a dimensional analysis approach. The results indicate that the strain gradient elasticity theory captures the size effect more sensitive in comparison with the nonlocal elasticity theory in bending analysis. In addition, modified boundary conditions in strain gradient elasticity theory can lead to converge the classical solution at large scales. To prove that the boundary conditions of nano-beam have the direct effect on mechanical behavior of structure, the sizedependent Young modulus of carbon nanotube (CNT) is investigated and the results show that the prediction of strain gradient elasticity theory with modified boundary conditions is in a good agreement with experimental results.

A Review on Stress and Deformation Analysis of Curved Beams under Large Deflection

“…Another extensively used higher order beam theory is the nonlocal continuum theory suggested by Eringen [42], which specifies the stress state at a given point as a function of the strain states at all points in the body. In recent years, several higher order beam models have been developed in the framework of micropolar elasticity theory [34], various strain gradient theories [32,33,37,[44][45][46][47]] and Eringen's nonlocal theory [48][49][50][51][52] to study static [34,[45][46][47]51], dynamic [31,32,37,45,49,50,52] and buckling [33,44,48] behavior of micro and nano size beam like structures.…”

“…Beam modeling based on higher order continuum theories, incorporating intrinsic size effects and surface effects, are amenable to numerical computations but their application is limited by the difficulties in calibration of the internal length scale and formulating the boundary conditions [47,51,54].…”

Weighted Residual Approach for Bending Analysis of Nanobeam Using by Modified Couple Stress Theory

“…A unified non-local formulation for bending, buckling and free vibration analysis of nanobeams published by Nikam and Sayyad [18]. Determination of the appropriate gradient elasticity theory for bending analysis of nanobeams by considering boundary conditions effect investigated by Shokrieh and Zibaei [19]. Beni [20] published study about size-dependent electromechanical bending, buckling and free vibration analysis of functionally graded piezoelectric nanobeams.…”

Nonlinear Vibration Analysis of Axially Functionally Graded Microbeams Based on Nonlinear Elastic Foundation Using Modified Couple Stress Theory

“…Experiments show that ignoring the internal length scale, which is the case in the classical continuum mechanics, in micro and nano structures can result in inaccurate structural predictions. To overcome this problem, a number of theories like couple stress theory and strain gradient theory [1][2][3][4][5] have been developed. In the couple stress theory, the effect of couple per unit area along with the effect of normal and shear forces is considered [6].…”