A new simple formulation of the orientation constraints for very flexible bodies is introduced in this investigation. This constraint formulation avoids using the tangent or cross-section frames previously used in the literature by directly using two position-gradient vectors to define the orientation constraint equations that eliminate the relative rotations between two bodies. The orientation constraints are commonly used to define clamped (rigid) joint between very flexible bodies, and rigid and less-flexible bodies. The very flexible bodies are modeled using the absolute nodal coordinate formulation (ANCF) which employs position gradients as nodal coordinates, while other rigid and less-flexible bodies have kinematics described in terms of orientation parameters. The less-flexible bodies can be modeled using the floating frame of reference formulation. Conventional clamped-end conditions eliminate all rigid body and deformation degrees of freedom because of the low-order of the finite element interpolation. When using a higher interpolation order, distinction is made between fully- and partially-clamped joints; the former eliminates all the degrees of freedom, while the latter eliminates only the relative translations and rotations, allowing for local deformations at the joint definition point. The singularity problem that arises, in the formulation of the orientation constraints, as the result of using ANCF orientation vectors derived using one position-gradient vector is demonstrated using simple, but common, example of a cantilever beam. The applicability of Saint–Venant principle, which states that the effect of the load diminishes away from the point of application of the load, to constraint forces that eliminate degrees of freedom is examined. Numerical results obtained in this study demonstrate that elimination of degrees of freedom, which leads to different kinematic structures and deformation basis-vectors, can be more significant in the case of soft materials.

在这项研究中引入了一种新的关于非常灵活的物体的方向约束的简单公式。这种约束的制剂避免了使用正切或横-截面帧通过直接使用两个位置梯度向量来定义的取向约束方程，消除了两个物体之间的相对旋转在以前的文献中使用。方向约束通常用于定义非常柔性的实体与刚性和较不柔性的实体之间的夹紧（刚性）接头。使用绝对节点坐标公式对非常柔软的物体进行建模（ANCF）使用位置梯度作为节点坐标，而其他刚体和不太灵活的物体具有根据方向参数描述的运动学。较不灵活的实体可以使用浮动参考框架建模公式。由于有限元插值的低阶，传统的夹紧端条件消除了所有刚体和变形的自由度。当使用较高的插补次数时，要区分完全夹紧接头和部分夹紧接头。前者消除了所有自由度，而后者则消除了相对的平移和旋转，从而允许在关节定义点处进行局部变形。使用简单但通用的悬臂梁示例演示了在定向约束条件的制定中出现的奇异问题，该问题是使用使用一个位置梯度矢量得出的ANCF定向矢量的结果。Saint-Venant原理的适用性，指出负载的影响从负载的施加点逐渐减小，限制消除自由度的力受到检查。在这项研究中获得的数值结果表明，消除自由度会导致不同的运动学结构和变形基础向量，对于软质材料可能更为重要。