A nonlinear microbeam model based on strain gradient elasticity theory with surface energy

Abstract

A microscale nonlinear Bernoulli-Euler beam model on the basis of strain gradient elasticity with surface energy is presented. The von Karman strain tensor is used to capture the effect of geometric nonlinearity. Governing equations of motion and boundary conditions are obtained using Hamilton's principle. In particular, the developed beam model is applicable for the nonlinear vibration analysis of microbeams. By employing a global Galerkin procedure, the ordinary differential equation corresponding to the first mode of nonlinear vibration for a simply supported microbeam is obtained. Numerical investigations show that in a microbeam having a thickness comparable with its material length scale parameter, the strain gradient effect on increasing the beam natural frequency is higher than that of the geometric nonlinearity. By increasing the beam thickness, the strain gradient effect decreases or even diminishes. In this case, geometric nonlinearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for beams with some specific thickness-to-length parameter ratios, both geometric nonlinearity and size effect have significant role on increasing the frequency of nonlinear vibration.

An Euler–Bernoulli second strain gradient beam theory for cantilever sensors

“…As Mindlin demonstrated that such a 3 theory is able to describe the surface tension for solids, this paper is intended to derive a tractable beam theory based on such a higher order material description. This contribution may therefore be seen as an extension of the recent propositions to describe the behavior of tiny Euler-Bernoulli [11][12][13] or Timoshenko [14] beams. The basic assumptions regarding the beam behavior are first presented.…”

A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory

“…It is not Later, Lazopoulos and Lazopoulos [17] used the same strain energy density and developed a linear Timoshenko beam model. Rajabi and Ramezani [18] used this strain energy to construct a non-linear Bernoulli-Euler beam model, and investigated the non-linear free vibration of microbeams.…”

Nonlinear Vibration Analysis of Microscale Functionally Graded Timoshenko Beams using the Most General form of Strain Gradient Elasticity

“…Ahmadi and his associates [34] studied the buckling of rectangular flexural microplates using the MCST. Based on the SGT with the inclusion of surface energy, Rajabi and Ramezani [35] developed a nonlinear Euler-Bernoulli beam model for the vibration of microbeams. Ansari et al [36] employed the SGT to study the nonlinear vibration of Timoshenko microbeams.…”