A new microstructure-dependent Saint–Venant torsion model based on a modified couple stress theory |
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| Authors: | GC Tsiatas JT Katsikadelis |
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| Institution: | 1. Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow 119991, Russian Federation;2. Southern Federal University, Faculty of Mathematics, Mechanics and Computer Science, Rostov-on-Don 344090, Russian Federation;3. Department of Applied Mathematics and Cybernetics, Tver State University, Tver 170100, Russian Federation;1. Department of Advanced Science & Technology, Toyota Technological Institute, 2-12-1 Hisakata, Tempaku-ku, Nagoya 468-8511, Japan;2. Department of Nanomechanics, Tohoku University, Aza Aoba 6-6-01, Aramaki, Aoba-ku, Sendai 980-8579, Japan;1. School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, NSW 2522, Australia;2. Department of Mechanical Engineering, McGill University, Montreal, Quebec H3A 0C3, Canada;1. School of Mechanical Engineering, University of Adelaide, South Australia, 5005, Australia;2. Department of Mechanical and Construction Engineering, Northumbria University, Newcastle Upon Tyne, NE1 8ST, UK;1. Vienna University of Technology, Institute for Mechanics of Materials and Structures, Karlsplatz 13, A-1040 Vienna, Austria;2. Department of Applied Mechanics and Mechatronics, IAMM FEI STU in Bratislava, Ilkovi?ova 3, 812 19 Bratislava, Slovakia;3. Tongji University, Siping Road 1239, Shanghai, China |
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| Abstract: | In this paper a new modified couple stress model is developed for the Saint–Venant torsion problem of micro-bars of arbitrary cross-section. The proposed model is derived from a modified couple stress theory and has only one material length scale parameter. Using a variational procedure the governing differential equation and the associated boundary conditions are derived in terms of the warping function. This is a fourth order partial differential equation representing the analog of a Kirchhoff plate having the shape of the cross-section and subjected to a uniform tensile membrane force with mixed Neumann boundary conditions. Since the fundamental solution of the equation is known, the problem could be solved using the direct Boundary Element Method (BEM). In this investigation, however, the Analog Equation Method (AEM) solution is applied and the results are cross checked using the Method of Fundamental Solutions (MFS). Several micro-bars of various cross-sections are analyzed to illustrate the applicability of the developed model and to reveal the differences between the current model and an existing one which, however, contains two additional constants related to the microstructure. Moreover, useful conclusions are drawn from the micron-scale torsional response of micro-bars, giving thus a better insight in the gradient elasticity approach of the deformable bodies. |
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