V – Problems in Magneto-solid Mechanics

Theory of electromagnetic elastic plates

In its simplest form the magnetoelastic buckling instability refers to the sudden bending transition of an elastic rod experiencing a uniform induction field applied at a normal angle with respect to its long axis. This fundamental physics phenomenon was initially documented in 1968, and, surprisingly, despite many refinements, a gap has always remained between the observations and the theoretical expectations. Here, we first renew the theory with a simple model based on the assumption that the magnetization follows the rod axis as soon as it bends. We demonstrate that the magnetoelastic buckling corresponds to a classical Landau second-order transition. Our model yields a solution for the critical field as well as the shape of the deformed rods which we compare with experiments on flexible ferromagnetic nickel rods at the centimeter scale. We also report this instability at the micrometer scale with specially designed rods made of nanoparticles. We characterized our samples by determining all of the relevant parameters (radius, length, Young modulus, magnetic susceptibility) and, using these values, we found that the theory fits extremely well the experimental results for both systems without any adjustable parameter. The superparamagnetic feature of the microrods also highlights the fact that ferromagnetic systems break the symmetry before the buckling. We propose a magnetic “stick–slip” model to explain this peculiar feature, which was visible in past reports but never detailed.

https://www.pnas.org/content/pnas/112/23/7135.full.pdf

A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods.

Active control of beam with magnetostrictive layer

Magnetorheological elastomers (MREs) are ferromagnetic particle impregnated rubbers whose mechanical properties are altered by the application of external magnetic fields. Due to their coupled magneto-mechanical response, MREs are finding an increasing number of engineering applications. One such application is in haptics, where the goal is to actively control surface roughness. One way to achieve this is by exploiting the unstable regime of MRE substrate/layer assemblies subjected to transverse magnetic fields. In this work, we study the response of such an assembly subjected to a transverse magnetic field and in-plane stress. The layer is made up of a transversely isotropic MRE material, whose energy density has been obtained experimentally, while the substrate is a non-magnetic isotropic pure polymer/gel. An analytical solution to this problem based on a general, finite strain, 2D continuum modeling for both the MRE layer and the substrate, shows that for adequately soft substrates there is a finite-wavelength buckling mode under a transverse magnetic field. Moreover, the critical magnetic field can be substantially reduced in the presence of a compressive stress of the assembly, thus opening the possibility for haptic applications operating under low magnetic fields.

https://hal-polytechnique.archives-ouvertes.fr/hal-00995717/document

Instability of a magnetoelastic layer resting on a non-magnetic substrate

The Euler buckling formula is a simple deduction from the Bernoulli-Euler theory of bending. Perhaps a correct magnetoelastic buckling formula with be obtained as a simple deduction from a correct theory of magnetoelastic bending. The present investigation is a first step toward creating such a theory.

Magnetostatic deformation experiment: Bending of a thin plate

Existing theoretical values and experimental values of magnetoelastic critical buckling fields for ferro- magnetic beams and plates differ by a factor of two. The usual theoretical treatment assumes that the plate (a) is immersed in an infinite uniform field, (b) is infinitely wide and long, and (c) is made of a linearly magnetic material. We present an experimental investigation, combined with a simple analysis, which determines specific criteria which must be satisfied by experimental programs in order to validate the three assumptions. It is shown that experimental arrangements which validate (a) and (c) are feasible, but not (b). Hence, there is need for improved theoretical treat- ments which take into account the finite size of the speci- mens.

Paper presents an investigation of the conditions which must be satisfied in an experimental program in order to check theoretical predictions of magnetoelastic buckling fields

M. O. Peach

Papers

Magnetostatic deformation experiment: Bending of a thin plate

Paper presents an investigation of the conditions which must be satisfied in an experimental program in order to check theoretical predictions of magnetoelastic buckling fields