A Cascading Mean-Field Approach to the Calculation of Magnetization Fields in Magnetoactive Elastomers

Abstract

We consider magnetoactive elastomer samples based on the elastic matrix and magnetizable particle inclusions. The application of an external magnetic field to such composite samples causes the magnetization of particles, which start to interact with each other. This interaction is determined by the magnetization field, generated not only by the external magnetic field but also by the magnetic fields arising in the surroundings of interacting particles. Due to the scale invariance of magnetic interactions (O(r−3) in d=3 dimensions), a comprehensive description of the local as well as of the global effects requires a knowledge about the magnetization fields within individual particles and in mesoscopic portions of the composite material. Accordingly, any precise calculation becomes technically infeasible for a specimen comprising billions of particles arranged within macroscopic sample boundaries. Here, we show a way out of this problem by presenting a greatly simplified, but accurate approximation approach for the computation of magnetization fields in the composite samples. Based on the dipole model to magnetic interactions, we introduce the cascading mean-field description of the magnetization field by separating it into three contributions on the micro-, meso-, and macroscale. It is revealed that the contributions are nested into each other, as in the Matryoshka’s toy. Such a description accompanied by an appropriate linearization scheme allows for an efficient and transparent analysis of magnetoactive elastomers under rather general conditions.

Field-Induced Transversely Isotropic Shear Response of Ellipsoidal Magnetoactive Elastomers

“…Accordingly, the complex interplay between the microstructure and macroscopic shape creates the necessity of a unified approach bridging different scales. For example, the cascading mean-field description of the magnetization field in MAE composites is recently proposed by splitting the field into three contributions on different scales [ 40 ]. As the magnetization field provides a detailed description of local and global effects, this strategy allows to decouple the short-range contributions from the long-range contributions to the magnetization field in the MAE.…”

“…The unified approach is based originally on the “ellipsoidal” approximation to the sample shape, though some limitations of this approximation have been recently reported [ 24 ]. Recent works provide the extension of unified approach to any arbitrary sample shapes [ 40 , 41 ].…”

Magneto-Mechanical Enhancement of Elastic Moduli in Magnetoactive Elastomers with Anisotropic Microstructures

“…Recently, an efficient approximation scheme [41,60] could be established to calculate magnetization fields in composite materials under rather general conditions. For example, a tensorial notation was introduced to describe the effects of arbitrarily oriented external magnetic fields and/or more generic particle microstructures or sample shapes.…”

“…Accordingly, the tensorial notation can be reduced to a scalar description with all fields oriented along the x-direction, i.e., H 0 = H 0 e 1 , M = M e 1 and consequently also the total magnetic field H = H e 1 (a unit vector e 1 is aligned along the x-direction). Assuming the linear magnetization regime, M = χ H with isotropic susceptibility χ, the average magnetization in the sample is found via the leading order approximation [41,60] as:…”

“…According to Refs. [53,60], the contribution due to microscopic particle structure is formally obtained as:…”