Bistable magnetomechanical behavior of ferromagnetic particles in an elastomer matrix
The magnetomechanical behavior of a pair of identical isotropic ferromagnetic particles embedded in an elastomer matrix is studied. In the absence of an external magnetic field particles stay in a multi-domain state so that their magnetic moments are virtually zero. As soon as the sample is subjected to the magnetic field, the latter induces magnetization of the particles thus entraining them in magnetostatic (ponderomotive) interaction. The arising forces (gradients of interparticle magnetic energy) tend to shift the particles from their initial locations and to drive them to positions that are more favorable from the magnetic energy viewpoint. However, any displacement of the particles induces elastic resistance forces in the matrix so that the equilibrium strain (the change of the interparticle distance) comes out as a balance between the magnetic and elastic forces. In the present paper, to obtain the results for real ferromagnet-elastomer composites, known as soft magnetic elastomers (SMEs), particles are assumed to magnetize nonlinearly, approaching a finite value (saturation) at high field. To quantitatively account for the above assumption, a conventional (Frölich-Kennelly) empirical magnetization law is adopted. This law is based on only two material parameters: initial susceptibility and saturation magnetization. For the effective elastic interaction between particles, a new analytical interpolation formula is proposed and verified against the numerical solution done using the finite-element method. A combination of magnetic (inherent to saturating particles) and elastic energy contributions yields the net energy function, minimization of which renders a set of possible equilibrium states of the two-particle system in question. It turns out that, as it happens with paramagnetic particles (linearly magnetizable), such a pair could display a magnetomechanical hysteresis: abrupt transformation of a remote pair into a tight cluster. However, contrary to the paramagnetic case, the range of initial interparticle distances, where this phenomenon takes place, is rather limited. On the other hand, a pair of ferromagnetic particles, unlike paramagnetic ones, admits a special type of magnetomechanical hysteresis, which we term “latent”. This means that, although the particle clustering cannot be initiated solely by an external field, whatever strong, the clustering could occur if the action of the field is combined with some non-magnetic stimulus striving to bring the particles closer, e.g. axial mechanical pressure. The obtained diagram of states for a particle pair is used for drawing estimates on the possibility of occurrence of magnetomechanical hysteresis in the ferromagnetic filler in a real SME under magnetization.