Ph.D. (Engg) : Investigations on Elastic Deformation of Ferromagnetic Rods and Ribbons

Bulk ferromagnetic materials exhibit Joule magnetostriction with characteristic strain magnitudes on the order of (10^{-6}) to (10^{-4}), which is often insufficient for the displacement requirements of modern soft robotic and adaptive structural systems. Ferromagnetic elastic slender structures provide a promising alternative, offering the potential for large actuation displacements under small external magnetic fields. This enhanced response results from a rich coupling between magnetic and elastic phenomena in slender structures, where external magnetic fields can induce significant displacements with relatively small field strengths. This thesis develops a novel unified theoretical framework to describe the coupled magnetoelastic behavior of ferromagnetic elastic rods and ribbons. The framework formulates the total energy functional, incorporating elastic and magnetic energy components for both soft and hard ferromagnetic materials. The elastic energy is derived from Kirchhoff and Wunderlich models for rods and ribbons. The magnetic energy is formulated using the micromagnetic energy functional composed of exchange, anisotropy, magnetostriction, demagnetization, and Zeeman energies. Central to the framework is the interplay between elastic energy and the competing magnetic effects: demagnetization energy in soft ferromagnets and Zeeman energy in hard ferromagnets.

In the first part of this thesis, we construct the total energy formulation for ferromagnetic rods undergoing planar deformation and utilize Kirchhoff kinetic analogy for our investigation. A detailed bifurcation analysis distinguishes the Hamiltonian phase portraits of elastic rods and soft ferromagnetic rods in longitudinal and transverse magnetic fields, revealing distinct subcritical and supercritical pitchfork bifurcations. The extension of Kirchhoff’s kinetic analogy to ferromagnetic rods enables the prediction of equilibrium shapes under various boundary conditions and applied fields. However, the kinetic analogy framework does not directly address the stability of these equilibrium states.

                                                                                                                                            Motivated by this limitation, the second part presents a comprehensive one-dimensional model for ferromagnetic elastic rods/ribbons systematically incorporating micromagnetic energy for curved geometries. The resulting equilibrium equations are derived for both soft and hard magnetic cases, which are then solved numerically to trace load–deflection responses. Stability analysis via a Sturm–Liouville eigenvalue approach reveals tensile critical buckling loads for soft ribbons and uncovers novel stable post-buckling configurations, especially for fixed-fixed boundary conditions under transverse magnetic fields. For hard ferromagnetic ribbons, the buckling loads are shifted, but the corresponding equilibrium shapes approach those of purely elastic ribbons. The restriction to planar deformations, however, leaves open the question of whether these configurations remain stable with respect to fully three-dimensional perturbations.

To address this issue, the third part extends the study to spatially deforming ferromagnetic elastic rods subjected to combined magnetic and terminal mechanical loading. The Hamiltonian derived from the total energy is analyzed, revealing a significant difference: the purely elastic and hard ferromagnetic rods exhibit subcritical pitchfork bifurcations, whereas the soft ferromagnetic rod shows no bifurcation under similar conditions. Furthermore, localized buckling deformation of soft ferromagnetic rod shows non-collinear straight segments.

This work is, to the best of our knowledge, the first to demonstrate the role of demagnetization energy in the large deformation of ferromagnetic slender structures. As one of the dominant contributions to the micromagnetic energy functional, the demagnetization energy is inherently geometry dependent and therefore crucial to structural deformation. As the slender structure deforms and its geometry changes and the demagnetization energy is correspondingly modified. Our results provide an understanding of the interplay of magnetic and elastic forces, paving the way for the design of advanced smart materials and potential applications in magnetically-actuated soft robots, adaptive medical devices, and remote actuation.