A Large‐Deformation Phase‐Field Approach for the Modeling of Magneto‐Sensitive Elastomers

Abstract

Magneto-sensitive materials show magneto-mechanical coupled response and are thus of increasing interest in the recent age of smart functional materials. Ferromagnetic particles suspended in an elastomeric matrix show realignment under the influence of an external applied field, in turn causing large deformations of the substrate material. The magneto-mechanical coupling in this case is governed by the magnetic properties of the inclusion and the mechancial properties of the matrix. The magnetic phenomenon in ferromagnetic materials is governed by the formation and evolution of domains on the micro scale. A better understanding of the behavior of these particles under the influence of an external applied field is required to accurately predict the behavior of such materials. In this context it is of particular importance to model the macro scopic magnetomechanically coupled behavior based on the micro-magnetic domain evolution. The key aspect of this work is to develop a large-deformation micro-magnetic model that can accurately capture the microscopic response of such materials. Rigorous exploitation of appropriate rate-type variational principles and consequent incremental variational principles directly give us field equations including the time evolution equation of the magnetization, which acts as the order parameter in our formulation. The theory presented here is the continuation of the work presented in [1,7] for small deformations. A summary of magneto-mechanical theories spanning over multiple scales has been presented in [4]. Primary variables in finite-deformation micro-magneto-elasticityThe macroscopic boundary-value-problem for a ferromagnetic body B embedded in free space Ω is a coupled multifield problem. Primary variables are the deformation map ϕ, the scalar magnetic potential φ and the reference magnetization director M given by,Note that the magnetization director is a priori defined only in the ferromagnetic material and has the geometric property M ∈ S d−1 , where S d−1 is the space of a unit sphere. The magnetization director and its rate in terms of the spin of magnetization ω ∈ R d are given as,This geometric property has to be consistently taken into account in our formulation. The magnetization of the material has a fixed value m s , where m s is the spontaneous magnetization of the material. The deformation gradient directly follows from the deformation map as F = Gradϕ and the reference self magnetic field as H = −Grad φ. Penalty formulation for the unity constraint on the directorAs mentioned before the geometric property of the magnetization director |M | = 1 has to be consistently taken into account in the formulation. A numerical implementation of the exact update of the magnetization is given in [7] and a staggered solution scheme is presented in [1]. A pure penalty formulation has been presented in [6], for small deformations. We propose here a staggered penalty solution scheme to satisfy this constraint. To this end the unity constraint on the magnetization director is relaxe...