One approach to explain Vibe I wanted to use within our papers from the beginning, was describing the effect as per the magnetoelastic energy balance. However, we agreed that it was just an approximation such that we axed it from our poster and publication. Magnetoelastic effects are often described by using a minimization of energy, in which the applied directions of each component of energy are expressed as angles. Introduction to Magnetic Materials by Cullity is a great source of information to teach these methods.
Elastic energy is often presented as elastic terms dependent on stress, σ , magnetostriction, γ. Magnetic energy is presented as applied field H, and saturation magnetization Ms. Anisotropy is effectively the ‘spring force’ acting to return the magnetization to its initial state, and is expressed as Ku.
As shown in the image, the stress direction for compressive or tensile forces from bending is indicated by the ‘X’ direction, applied circumferential field as induced by current in the ‘Y’ direction, crystal anisotropy will vary depending upon the orientation of any particular crystal, and the resulting magnetization is represented by Ms. In order to solve for a result, the distribution of each crystal dependent upon the structure of the material needs to be considered. To carry this out, Ku will be discretized and solved for independently over the full expected distribution. For the material we were using, we will solve for the distribution of Ku from 0 to π.
An example is shown in the figure below. In A), there is no applied field or stress, in which the magnetization is represented by a uniform distribution from 0 to π . In B), a stress is applied, which the magnetostriction causes the magnetization to rotate toward the direction of applied stress. In C), only a field H is applied, causing the magnetization to rotate toward the direction of applied field. In D), a field and stress are applied. It can be noted the applied field causes the magnetization to rotate away from direction of applied stress by comparing this figure to B). Figure E) is indicating compressive rather than tensile stress, in which F) is compressive stress combined with applied field H.
