Analysis of planar arbitrarily curved microbeams with simplified strain gradient theory and Timoshenko–Ehrenfest beam model

Author:

Vo Duy1ORCID,Aung Zwe Yan1,Le Toan Minh2,Suttakul Pana3,Atroshchenko Elena4,Rungamornrat Jaroon1ORCID

Affiliation:

1. Center of Excellence in Applied Mechanics and Structures, Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok, Thailand

2. Center of Excellence in Applied Mechanics and Structures, Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok, Thailand; Department of Civil Engineering, University of Calabria, Rende, Italy

3. Department of Mechanical Engineering, Faculty of Engineering, Chiang Mai University, Chiang Mai, Thailand

4. School of Civil and Environmental Engineering, University of New South Wales, Sydney, NSW, Australia

Abstract

As the first endeavor in the context of Mindlin’s strain gradient theory, this study contributes a systematic and rigorous derivation for governing equations and boundary conditions of planar arbitrarily curved microbeams. The Timoshenko–Ehrenfest beam model is incorporated into a simplified version of Mindlin’s strain gradient theory. Kinematic unknowns include displacement components of the beam axis in the local coordinate system and the rotation of cross-section. Since the derived governing equations and boundary conditions are extremely complex, analytical solutions are not achievable for microbeams having non-uniform curvature. To facilitate the numerical analysis, two isogeometric collocation formulations are proposed, that is, displacement-based and mixed formulations. Several tests are designed to evaluate the accuracy and reliability of the proposed isogeometric collocation formulations, especially with respect to the well-known locking pathology. It is found that the mixed formulation is more accurate and robust than the displacement-based one. Therefore, the mixed formulation is then used to numerically investigate the size-dependent behavior and stiffening effect. Furthermore, some informative tests are performed to delineate the significance of the curviness in the prediction of structural responses of planar arbitrarily curved microbeams, which appears to be still an unanswered issue.

Funder

Chiang Mai University

Chulalongkorn University

Publisher

SAGE Publications

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