Mathematical model of a three-layer micro- and nano-beams based on the hypotheses of the Grigolyuk–Chulkov and the modified couple stress theory

The flapwise bending free vibration analysis of isotropic rotating Timoshenko microbeams, including the size effects, is presented in this paper. A nonclassical theory, i.e. the Modified Couple Str...

The Flapwise Bending Free Vibration Analysis of Micro-rotating Timoshenko Beams Using the Differential Transform Method:

On the strain gradient elasticity theory for isotropic materials

Abstract New models for plane curved rods based on linear couple stress theory of elasticity have been developed.2-D theory is developed from general 2-D equations of linear couple stress elasticity using a special curvilinear system of coordinates related to the middle line of the rod as well as special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and rotation along with body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby, all equations of elasticity including Hooke’s law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear couple stress theory of elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scales when taking into account couple stress and rotation effects.

Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

In this paper, the transverse vibration of cracked nano-beam has been studied based on modified couple stress theory. Crack is modeled by a rotational spring that creates a discontinuity. Frequency equations of cracked nano-beam with some typical boundary conditions are derived and the frequency parameter for different crack positions, crack parameter and length scale parameters value are calculated. According to the results, it can be found that the effects of the crack parameter and crack location on transverse frequency of the cracked nano-beam are significant. In addition, scale effect parameter is one of the important parameter in nanoscale that must be obtained. Finally, the results of special cases show good agreement with the results available in the literature.

Size Effect on Free Transverse Vibration of Cracked Nano-beams using Couple Stress Theory

In this paper, a size-dependent formulation for the Bernoulli-Euler beam is developed based on a new model of couple stress theory presented by Hadjesfandiar i and Dargush. The constitutive equation obtained in this new model, consists of only one length scale parameter that is capable of capturing the micro-structural size effect in predicting the mechanical behavior of the structure. Having one length scale parameter is claimed to be an advantage of the model in comparison with the classical couple stress theory. The governing equations and boundary conditions of the Bernoulli-Euler beam are developed using the variational formulation and the Hamilton principle. The static bending and free vibration problems of a Bernoulli-Euler beam with various boundary conditions are solved. Numerical results demonstrate that the value of deflection predicted by the new model is lower than that of the classical theory. It is also found that natural frequencies obtained by the present couple stress model are higher than those predicted by the classical theory. The differences between results obtained by the present model and the classical theory become significant as the thickness of the beam gets close to the length scale parameter of the beam material

https://www.ije.ir/article_72328_2b70ef61b0ba12b60a8b15bfe8760c4e.pdf

A. H. Abolghasemi

Papers

A Size-dependent Bernoulli-Euler Beam Formulation based on a New Model of Couple Stress Theory