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On a nonlinear theory of thin rods
Institution:1. Department of Applied Mathematics, University of Twente, Netherlands;2. LabMath-Indonesia, Bandung, Indonesia;3. Department of Mathematics, Institut Teknologi Bandung, Indonesia;1. Department of Mechanical Engineering, Northwestern University, Evanston, Illinois, 60208, USA;2. Department of Civil Engineering, Faculty of Engineering, Ariel University, Ariel 40700, Israel;3. School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Ramat Aviv 6997801, Israel
Abstract:In this paper, a nonlinear theory for a straight rod is presented from the general theory of the three-dimensional deformable-body in the Cartesian coordinate frame. A set of nonlinear strains is presented, and the stretch on central curve exactly satisfies the deformation geometrical relations. The relations between the Euler angles and deformation are given from the curvatures and torsion curvatures of the central curves, which can easily explain the existing theories of rods and beams. Full nonlinear equations of motion for a nonlinear rod are developed via the vector form. Such a treatise is different from the traditional treatises of nonlinear rods, based on the Cosserat’s theory (e.g., Cosserat and Cosserat 1] in 1896) or the Kirchhoff assumptions (e.g., Kirchhoff 18] in 1859; Love 3] in 1944). This paper extends the ideas of Galerkin 4] in 1915. The nonlinear theory of thin rods can reduce to the existing theories for thin rods and beams, such ideas presented in this paper can be applied for development of the nonlinear theory for plates and shells as well.
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