We present in this contribution an interpolation of the rotations describing the Cosserat frame for a Kirchhoff rod. The frames are characterized by the director perpendicular to the cross section being also aligned with the tangential direction of the rod’s axis. The formulation is based on a G1 Hermite interpolation of the axis, described by the position of the rod itself but also with the above tangential direction to it. In this way, the second derivative of the axis position can be easily obtained, as needed in these rods without any transverse shear strain. Orthogonal to this tangential director are the other two directors corresponding to the cross section of the rod. The new interpolation of the rotation considers two parts. We first begin with an intermediate interpolation for a frame based on the slerp function (the so-called spherical linearized interpolation), and then a secondary rotation that leads the director perpendicular to the cross section to align with the tangential direction previously calculated of the rod’s axis. All this is given in an exact closed-form expressions. The formulation is then analyzed in terms of their invariant properties, in both the nonlinear and linearized forms. In this setting, the formulation is then shown that it is invariant, and that the nodal forces and moments do satisfy their equilibrium relations. Several numerical examples are presented showing these properties, as well as the improvement gained with other formulations not showing those orthonormality of the Cosserat frame in the full three-dimensional range.

中文翻译：

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用于非线性三维基尔霍夫杆的 Cosserat 导向器的新有限元插值


在这篇文章中，我们提出了一个旋转插值，描述了基尔霍夫杆的 Cosserat 框架。框架的特点是垂直于横截面的导向器也与杆轴的切向对齐。该公式基于轴的 G1 Hermite 插值，由杆本身的位置描述，但也由上述切向方向描述。通过这种方式，在这些杆中可以很容易地获得轴位置的二阶导数，而不需要任何横向剪切应变。与这个切向导向器正交的是对应于杆横截面的另外两个导向器。旋转的新插值考虑了两个部分。我们首先从基于 slerp 函数的帧的中间插值（所谓的球形线性插值）开始，然后是二次旋转，使导向器垂直于横截面，与先前计算的杆轴的切向对齐。所有这些都以精确的封闭式表达式给出。然后根据非线性和线性化形式的不变性质分析公式。在这种情况下，公式表明它是不变的，并且节点力和弯矩确实满足它们的平衡关系。给出了几个数值示例，显示了这些特性，以及使用其他公式获得的改进，这些公式在整个三维范围内没有显示 Cosserat 框架的正交性。