Relative orientation constraints in the nonlinear large displacement analysis: application to soft materials

Abstract

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.