Transient analysis of Cosserat rod with inextensibility and unshearability constraints using the least-squares finite element model

• The complete dynamic equations for the transient response of Cosserat rod having inextensibility and unshearability constraints are derived.
• Least-squares finite element formulation of the governing equations is presented.
• Solution of the resulting non-linear equations using Newton׳s method is outlined.
• Numerical examples of the dynamic response due to impulse force and time-dependent follower force are presented.

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 surgical thread 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 α-family of time derivarive 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 non-linear finite element equations. Newton׳s method is utilized to solve the non-linear finite element equations. Dynamic response due to impulse force and time-dependent follower force at the free end of the rod is presented as numerical examples.