空間曲線棒の有限変形理論

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  • ON THE NONLINEAR THEORY OF A SPACE CURVED ROD
  • クウカン キョクセンボウ ノ ユウゲン ヘンケイ リロン

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The aims of this paper are to formulate the rigorous governing equation for the problem of large displacements and large rotations of an elastic rod, whose axial curve is a space curve, in the reference state and further to represent the various approximate equations of the derived rigorous governing equation by means of approximations without loosing generality for practical uses. It is assumed from the viewpoint grasping the main deformation behaviour that the displacement function of the rod consists of the plan displacement composed of stretching, bending, the transverse shear deformation and the deformation of the cross section without the local deformation and of warping occurring by twisting, and that the latter warping is addition to in the displacement state of the former. Wherein it is also assumed that warping is expressed by the product of warping parameter and the modified warping function considered the effect of the initial curvature and torsion of the axial curve in the St. Venant's warping function. Under such assumptions to the displacement functions, the governing equation of elastic rods is obtained by reducing the three-dimensional body to the one-dimensional one through the modified Hellinger-Reissner's variational principle. Furthermore, the simplified equations of the derived rigorous governing equation are presented by means of the following assumptions. (i) the assumption of the thinness of a rod, (ii) the neglect of the effects of the curvature and torsion in warping, (iii) the neglect of the nonlinear terms involving warping, (iv) the assumption of the rigid displacement of the cross section, (v) small rotation and the neglect of the warping components α^2, Although these assumptions can use arbitrarily and independently, the simplified equations in this paper are developed by means of the above assumptions in turn. The results derived here contain the theories of Washizu and Wempener and further the classical result.

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