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Free vibration of a hyper-elastic microbeam using a new “augmented Biderman model”

Karami Mohammadi Ardeshir, Danaee Barforooshi Saeed

Journal of Theoretical and Applied Mechanics

2019

Vol. 57 nr 3

739--750

A new augmented Biderman model inspired by the modified couple stress theory has been introduced to investigate the size effect in addition to nonlinear material effects. Then, this model is used to investigate free vibration of a hyper-elastic microbeam. Classical Biderman strain energy does not include the effect of small size in hyper-elastic materials. In order to consider the effect of small size, terms inspired by the modified couple stress theory are added to the classical Biderman strain energy function. In order to provide the possibility of calculating these terms, a relation between the material constants in the hyper-elastic Biderman model and the linear elastic constants is obtained. The equations of motion of the microbeam is obtained based on the extended Hamilton principle, and then is solved using Galerkin discretization and perturbation methods. The effect of thickness to length scale ratio on the normalized frequency is studied for different modes. It is shown that when thickness gets larger in comparison with the length scale parameter, the normalized frequency tends to classical Biderman results. The results obtained are validated by results of the Runge-Kutta numerical method and indicate an excellent agreement. Mode shapes of the microbeam based on the classical and the augmented models are depicted, where the augmented model anticipates stiffer behavior for hyperelastic microbeams.