ON THE DYNAMICS OF RODS IN THE THEORY OF KIRCHHOFF AND CLEBSCH
ON THE DYNAMICS OF RODS IN THE THEORY OF KIRCHHOFF AND CLEBSCH
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DOI：
10.1007/bf00375625
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发表时间：
1993-01-01
影响因子：
2.5
通讯作者：
TOBIAS, I
中科院分区：
文献类型：
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作者：
COLEMAN, BD;DILL, EH;TOBIAS, I
我们在这里讨论由 Kmcm-ioFF [1, 2] 和 CL~ Bsc~[3, 4] 产生的弹性杆理论的动力学方程。这种适当不变的理论适用于尽管旋转可能很大，但相对于未扭曲结构的应变仍然很小的运动。它被构造为一阶理论，即在厚度、曲率、扭曲和延伸的适当无量纲测量中完整到二阶误差内的理论。在细杆的一阶理论中，人们可以将杆视为不可延伸，我们一开始就这样做。因此，在每个时间t，采用轴向曲线的弧长参数s~(t)作为材料坐标，即在材料点处的值随时间恒定的参数，不仅横截面上的剪切力的合力，而且杆中的拉力都是本构方程无法给出的反作用量。考虑一下自然棱柱形和动态对称的杆，即，在未扭曲的无应力配置中是圆柱体的杆，其准线虽然不一定是圆，但以相等的主惯性矩界定图形。如果杆的运动是平面且无扭转的，即纯弯曲运动，如果它始终位于包含每个横截面的惯性主轴的固定平面内。对于这样的运动，我们在~上采用固定的笛卡尔坐标系，并且，为了与本文稍后给出的更一般运动的讨论保持一致，我们将横坐标称为z，纵坐标称为x。我们可以将弧长坐标为 s 的截面上 t 时刻合力 F 的 z 和 x 分量写为 FZ (s, t)、FX (s, t)。杆的运动可以通过给出(t)上的点的(z，x)坐标作为s和t的函数来描述。设 O (s, t) 从 z 轴到 s 处的切线的逆时针角度，我们有
We discuss here the dynamical equations of a theory of elastic rods that is due to Kmcm-ioFF [1, 2] and CL~ Bsc~[3, 4]. This properly invariant theory is applicable to motions in which the strains relative to an undistorted configuration remain small, although rotations may be large. It is constructed to be a first-order theory, ie, a theory that is complete to within an error of order two in an appropriate dimensionless measure of thickness, curvature, twist, and extension.In a first-order theory of thin rods, one can treat the rod as inextensible, and we do so here at the outset. Thus, at each time t, the arc-length parameter s for the axial curve~(t) is employed as a material coordinate, ie, a parameter whose value at a material point is constant in time, and not only the resultant of the shearing forces on a cross section, but also the tension in the rod, are reactive quantities not given by constitutive equations. Consider for a moment a rod that is naturally prismatic and dynamically symmetric, ie, a rod that in an undistorted stress-free configuration is a cylinder whose directrix, although not necessarily a circle, bounds a figure with equal principal moments of inertia. A motion of the rod is said to be planar and twist-flee, ie,, a motion of pure flexure, if it is such that~(t) lies at all times in a fixed plane~ which contains a principal axis of inertia of each cross section. For such a motion we employ a fixed Cartesian coordinate system on~, and, for consistency with a discussion of more general motions to be given later in this paper, we call the abscissa z and the ordinate x. We may write FZ (s, t), FX (s, t) for the z-and x-components of the resultant force F at time t on the cross section with arc-length coordinate s. The motion of the rod may be described by giving the (z, x)-coordinates of the points on~(t) as functions of s and t. With O (s, t) the counterclockwise angle from the z-axis to the tangent of~(t) at s, we have