Abstract:

In this study,  time-dependent fully discretized least-squares finite element model is developed for the transient response of Cosserat rod having inextensibility and unshearability constraints to simulate a thin rod in space. Starting from the kinematics of the rod for large deformation, the linear and angular momentum equations along with constraint conditions for the sake of completeness are derived. Then, the alpha-family of time approximation is used to reduce the governing equations of motion to obtain a semi-discretized system of equations, which are then fully discretized using the least-squares approach to obtain the nonlinear finite element equations. Newton's method is utilized to solve the nonlinear finite element equations. Dynamic response due to impulse force and time-dependent follower force at the free end of the rod are presented as numerical examples.

Numerical examples:

Simulation of two transient analyses of a thin Cosserat rod are shown in the following video considering the linear material model.

                   Response of a rod under impulsive force                                                       Response of a rod due to follower force

1. Arbind, A., and J. N. Reddy. "Transient analysis of Cosserat rod with inextensibility and unshearability constraints using the least-squares finite element model." International Journal of Non-Linear Mechanics 79 (2016): 38-47.