Unified static equilibrium modeling and analysis of elastic rods with large deformations for complex constraints

Abstract

This paper presents a unified framework to describe the static equilibrium modeling and analysis of elastic rods with large deformations. Through an innovative combination of the Hamiltonian principle in arc coordinates and nonlinear optimization based on the principle of minimum potential energy, our method enables us to support the nonlinear statics model and deal with the different complex constraints of an elastic rod, including the end restraint, self-contact, through-hole, and surface contact Based on the idea of dynamic analogy, the equilibrium shape of an elastic rod is described by the movement and rotation of the moving coordinate system along an arc relative to the static coordinate system. Euler parameters are used as generalized variables to describe the strain energy and the external force potential energy of the elastic rod. Taking the total potential energy of the elastic rod as the objective function and different contact forms as the constraint conditions, a nonlinear optimization model is introduced to describe the equilibrium shape of the elastic rod. Based on a primal–dual interior-point method, the model is solved numerically. Considering various constraints and initial conditions, the geometric shape of the elastic rod is simulated. The simulation results show that the unified model proposed in this paper can effectively deal with the static equilibrium problem of elastic rods with large deformations for multiple complex constraints.