Euler elastica as a Γ-limit of discrete bending energies of one-dimensional chains of atoms

In the 1920s, Hencky proposed a discrete elastica model describing a chain of identical rigid bars connected by torsional springs. Hencky observed that this discrete elastica model converges to Euler’s elastica as the number of bars increases while their lengths decrease, and Hencky’s bar-chain model has been used primarily as an approximation of Euler’s elastica. A Hencky-type bar-chain model can also be incorporated into a Frenkel–Kontorova-type discrete atomistic model, where the joints and bars represent the atoms and interatomic bonds, respectively, while the entire chain of atoms interacts with either a substrate or other chains. The energy of a continuum system corresponding to this Frenkel–Kontorova-type model can then be recovered by taking an appropriate discrete-to-continuum limit. Developing a correct limiting procedure for the discrete elastica establishes the bending component of this continuum energy. In this paper we use Γ-convergence to rigorously show that as the bar length in the discrete elastica model we consider goes to 0, the bending energies of the chain Γ-converge to the continuum bending energy associated with Euler’s elastica.

Magneto-elastic buckling of an Euler beam

We consider nonlinear elastic deformations of a magneto-elastic beam, using a combined experimental and theoretical approach. In the experiments, a beam had one end clamped with a magnet attached at its free end. When it was placed in an external magnetic field, it was susceptible to Euler beam buckling. However, the classic supercritical bifurcation associated with this buckling became subcritical when an attracting magnet was introduced in close proximity to the beam. To understand these experiments, we develop a model that couples the Euler elastica and dipole magnetic interactions with a uniform external field. The numerical model captures the observed behaviour well and shows that the supercritical magnetic field strength depends almost exclusively on elastic properties of the beam and strength of the permanent magnet, whereas the subcritical behaviour also depends on the separation distance between the attracting pair of magnets. We examine the bifurcation behaviour of the nonlinear system and show that for sufficiently small inter-magnet separation distances, other buckled states coexist with the fundamental mode.