V – Problems in Magneto-solid Mechanics

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.

Of interest here is the stability of a rectangular block subjected to a uniform magnetic field perpendicular to its longitudinal axis. The two ends of the block are frictionless and kept parallel to each other. This boundary value problem is motivated by the classical problem of magnetoelastic buckling in which a cantilever beam subjected to a transverse magnetic field buckles when the applied field reaches a critical value. This work presents a finite strain continuum mechanics formulation of the stability problem of a homogeneous, compressible, magnetoelastic rectangular block in plane strain subjected to a uniform transverse magnetic field. The applied variational approach employs an unconstrained energy minimization recently proposed by the authors. The analytical solution for the critical buckling fields for both the antisymmetric and symmetric modes are obtained for three different constitutive laws. The corresponding result for thin beams is extracted asymptotically for a special material and the solution is compared to previously published results. The critical magnetic field is shown to increase monotonically with the block's aspect ratio for each material and mode type. Antisymmetric modes are always the critical buckling modes for stress saturated and neo-Hookean materials, except for a narrow range of moderate aspect ratios (about 0.25 ) where symmetric modes become critical. For strain-saturated solids no buckling is possible above a maximum aspect ratio.

Magnetoelastic buckling of a rectangular block in plane strain

In order to simulate the experimental phenomenon of increase of natural frequency to a cantilevered ferromagnetic beam plate in in-plane magnetic fields, a theoretical model for behaving the magnetoelastic interaction is proposed in this paper based on the variational principle of energy functional of the system. It is found that the expression of magnetic force is distinctly different from those of the existing theoretical models in publications, and the experimental phenomenon is successfully simulated by this theoretical model. After the increase of natural frequency is quantitatively considered in the predictions of magnetic damping, the theoretical predictions of magnetic damping ratio are agreement with the corresponding experimental data well.

A Theoretical Prediction of Natural Frequency of a Ferromagnetic Beam Plate With Low Susceptibility in an In-Plane Magnetic Field

As an essential model of magnetoelastic interaction between magnetic field and mechanical deformation, the study on magnetoelastic buckling phenomenon of soft ferromagnetic plates in a magnetic environment has been conducted. One of the key steps for the theoretical prediction of the critical magnetic field is how to formulate magnetic force exerted on the magnetized medium. Till today, the theoretical predictions, from theoretical models in publications, of the magnetoelastic buckling of ferromagnetic cantilevered beam-plate in transverse magnetic field are all higher than their experimental data. Sometimes, the discrepancy between them is as high as 100%. In this paper, the macroscope formulation of the magnetic forces is strictly obtained from the microscope Amperion current model. After that, a new theoretical model is established to describe the magnetoelastic buckling phenomenon of ferromagnetic thin plates with geometrically nonlinear deformation in a nonuniform transverse magnetic field. The numerical method for quantitative analysis is employed by combining the finite elemental method for magnetic fields and the finite difference method for deformation of plates. The numerical results obtained from this new theoretical model show that the theoretical predictions of critical values of the buckling magnetic field for the ferromagnetic cantilevered beam-plate are in excellent agreement with their experimental data. By the way, the region of applicability to the Moon-Pao's model, or the couple model, is checked by quantitative results.

A theoretical model of magnetoelastic bucking for soft ferromagnetic thin plates

Magnetoelastic Buckling of Thin Magnetically Soft Plates in Cylindrical Mode

This paper does not stand alone; it is directly related to N.S. Christopherson's experimental study1,2 of the magnetoelastic bending of thin steel plates, presented at a recent SEM meeting. It is, in fact, an extension of that study and relies upon some of Christopherson's data.

Magnetoelastic buckling: why theory and experiment disagree

Existing theoretical values and experimental values of magnetoelastic critical buckling fields for ferromagnetic 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 treatments which take into account the finite size of the specimens.

Magnetoelastic buckling: Theory vs. experiment: 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

Edge Effect Influence on Magnetoelastic Buckling of Rectangular Plates

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

G. L. Viegelahn

Papers

Magnetoelastic Buckling of Thin Magnetically Soft Plates in Cylindrical Mode

Magnetoelastic buckling: why theory and experiment disagree

Magnetoelastic buckling: Theory vs. experiment: 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

Edge Effect Influence on Magnetoelastic Buckling of Rectangular Plates

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