Bonded-particle model for magneto-elastic rods

Abstract

We develop a bonded-particle model for magneto-elastic rods that unifies large deformations, contact, and long-range magnetic interactions within a single discrete-element framework. The rod is discretized into orientable particles connected by co-rotational bonds that capture stretching, shearing, bending, and twisting through a symmetric decomposition of relative displacement and rotation. Magnetic coupling is introduced at the particle level: each particle carries a dipole moment that rotates with it, enabling both external-field actuation and long-range dipole--dipole interactions without modifying the structural formulation. We implement the model in LAMMPS to take advantage of its parallel efficiency, long-range electrostatic solvers, and multiphysics capabilities. We validate the framework against three benchmark problems: writhing instabilities of straight and curved rods under extreme twisting, large deflections of magnetized beams in uniform and constant-gradient fields, and mechanical hysteresis of helical rods with dipole--dipole interactions. To demonstrate multiphysics capability, we couple the model with a lattice Boltzmann fluid solver via the immersed boundary method and simulate filaments in oscillatory channel flow and fluid pumping by magnetically actuated cilia arrays. Across all examples, the model shows good agreement with experimental, analytical, and numerical reference results.

Figures (10)

• Figure 1: Illustration of the bonded-particle model for magnetic rods. (a) Each particle is treated as a magnetic dipole. An external magnetic field or dipole--dipole interaction exerts a force and torque on the particle. Neighboring particles are connected by an elastic bond. (b) The relative displacement and rotation between two bonded particles are decomposed into four linear spring interactions. The reference configurations are displayed below the corresponding deformed configurations.
• Figure 2: (a) The reference (light) and deformed (dark) bond. (b) The body frames of the two particles are defined by the quaternions $q_1$ and $q_2$ in the global frame. The central frame, $C$, is defined by $q_C$, which is the average of $q_1$ and $q_2$. (c) The relative displacement is measured from the moving $C$ frame. (d) The $C'$ frame is obtained by aligning the $C$ frame to the bond vector. (e) The orientation of particles $1$ and $2$ in the $C'$ frame is defined by the quaternions $u^\ast$ and $u$, respectively. Both quaternions are then decomposed in a bend before twist order such that the total rotation of particle $2$ with respect to particle $1$ is a sequence of four rotations. (f) Illustration of the swing--twist decomposition.
• Figure 3: The setup for the twisting experiment of lazarus2013continuation. (a) An initially straight rod is compressed by a fixed amount causing it to buckle under its own weight. The rod is then twisted multiple times and the resulting shape is studied. (b) An ideal naturally curved rod coiled into a circle with curvature $\kappa$ can be straightened by applying a pure end moment $M = \kappa E I$.
• Figure 4: Extreme twisting of a pre-buckled initially straight rod. (a) Top view snapshots at different twisting angles $\Phi$ comparing the experimental images of lazarus2013continuation and ours. (b) The maximum deflection as a function of the twist angle. The red points are their experimental measurements and the green curve is their semi-analytical solution, while the blue curve is our simulation result. The simulated plectoneme (inset) formation happens near their predicted critical angle $\theta_c = 11.33\pi$.
• Figure 5: Extreme twisting of a pre-buckled, naturally curved rod. The rod is coiled in its stress-free state and straightened before undergoing the same procedure as in Fig. \ref{['sn-article:fig:heavy-straight-rod']}. (a) Top view snapshots at different twisting angles comparing the experimental images of lazarus2013continuation with ours. (b) Maximum deflection as a function of twist angle. The red points are their experimental measurements and the green curve is their semi-analytical solution, while the blue curve is our simulation result. The plectoneme forms at a critical angle $\theta_c = 16.15 \pi$.