Size effects on shear deformable beam structures

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Journal Title
Journal ISSN
Volume Title
School of Engineering | Doctoral thesis (article-based) | Defence date: 2017-10-27
Date
2017
Major/Subject
Mcode
Degree programme
Language
en
Pages
38 + app. 76
Series
Aalto University publication series DOCTORAL DISSERTATIONS, 118/2017
Abstract
Bending, buckling and free vibration of isotropic as well as anisotropic centrosymmetric beams are formulated and studied within strain and velocity gradient theory. Different beam theories such as the third-order shear deformable, Timoshenko and Euler-Bernoulli beam theories are considered. A variational approach is applied in order to determine the equilibrium equations together with the classical and nonclassical boundary conditions. For studying the buckling of the beams, the nonlinear von Kármán strain tensor is considered in order to investigate the case of small strains but moderate rotations. The general formulations derived for the anisotropic beams can be easily simplified for more common cases of anisotropy such as orthotropy and transverse isotropy. The analysis of size effect on anisotropic beams is missing in the literature, while the present work enables one to investigate the size effect on ultra-small anisotropic centrosymmetric beam structures. The micro- or nanoscale beams are broadly used in sensors, resonators, actuators, nano- and microelectromechanical systems. In order to investigate the effect of isotropic or anisotropic length scale parameters on the static or dynamic behaviour of the beams, an analytical solution is provided for each case and the results are illustrated in figures. It is observed that the internal length scale parameters in different directions have a great impact on the behaviour of the beams, when the size of the structure is not negligible. This fact emphasizes the importance of considering the size effect in designing the miniature structures. Since the analytical solution is not available for all types of boundary conditions, a numerical method based on isogeometric analysis is developed. The comparison of the results obtained by the two different methods show that the numerical method works properly. Finally, a 3-D formulation for flexoelectric anisotropic materials within strain gradient elasticity is provided briefly. The simplification of the 3-D formulation for specific structures such as beams is the subject of future work.
Description
Supervising professor
Paavola, Juha, Prof., Aalto University, Department of Civil Engineering, Finland
Thesis advisor
Mousavi, S. Mahmoud, Dr., Karlstad University, Sweden
Keywords
beam, strain gradient, shear deformable, variational approach
Other note
Parts
  • [Publication 1]: Saba Tahaei Yaghoubi, S. Mahmoud Mousavi, Juha Paavola. Strain and velocity gradient theory for higher-order shear deformable beams. Archive of Applied Mechanics, 85:877-892, July 2015.
    DOI: 10.1007/s00419-015-0997-4 View at publisher
  • [Publication 2]: Saba Tahaei Yaghoubi, S. Mahmoud Mousavi, Juha Paavola. Size effects on centrosymmetric anisotropic shear deformable beam structures. ZAMM-Journal of Applied Mathematics and Mechanics/ Zeitschrift für Angewandte Mathematik und Mechanik, 97:586-601, May 2017.
    DOI: 10.1002/zamm.201600153 View at publisher
  • [Publication 3]: Saba Tahaei Yaghoubi, S. Mahmoud Mousavi, Juha Paavola. Buckling of centrosymmetric anisotropic beam structures within strain gradient elasticity. International Journal of Solids and Structures, 109:84-92, 15 March 2017.
    DOI: 10.1016/j.ijsolstr.2017.01.009 View at publisher
  • [Publication 4]: Saba Tahaei Yaghoubi, Viacheslav Balobanov, S. Mahmoud Mousavi, Jarkko Niiranen. Variational formulations and isogeometric analysis for the dynamics of anisotropic gradient-elastic flexible and shear deformable beams. European Journal of Mechanics-A/Solids, Under revision, 21 pages, 2017
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