Magnetoelasticity and Soft Actuators

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Motivation

The prospect of materials that change their mechanical behavior in response to an external magnetic field opens a route to soft, lightweight actuators that can be controlled remotely and without contact. This makes them attractive for robotics, biomedical devices, and micro-electromechanical systems. The physics is rich: a magnetic field couples to both the magnetization of the material and its elastic deformation, giving rise to nonlinear, multi-field problems that resist purely numerical treatment and demand the development of reduced, analytically tractable models.

My work in this area spans two decades and has evolved from the classical theory of rigid ferromagnets, through the thermodynamics of shape-memory alloys and the mechanics of magneto-rheological elastomers (MREs), to recent questions in homogenization, rigorous dimension reduction, and the variational foundations of magnetostatics. A recurring aim is to understand not only how magnetic bodies deform, but also which mathematical description of the magnetic state makes their equilibrium transparent.

Details

Rigid ferromagnets and domain-wall dynamics (2001–2006)

The early work set the theoretical stage by studying idealized models of hard ferromagnets, where the key mechanical quantity is the velocity of magnetic domain walls. With P. Podio-Guidugli, a continuum model for domain-wall motion was formulated and analyzed in the quasi-static and dynamic regimes:

These results established the importance of carefully accounting for dissipative mechanisms in magnetized media.

Thermodynamics of ferromagnets and shape-memory alloys (2009–2013)

The next thread addressed the interaction between magnetism, heat conduction, and solid-state phase transitions — the setting of magneto-mechanical shape-memory alloys (SMAs), which change both shape and magnetic anisotropy during martensitic transformation.

Magneto-rheological elastomers: large-deformation rod and beam theories

The focus then shifted to soft magneto-active materials — MREs, in which hard magnetic particles are embedded in a silicone matrix. The hallmark of these composites is that their elastic modulus is low enough that moderate magnetic fields produce large deflections. To exploit this for actuation, one needs structurally reduced models that are both physically accurate and computationally tractable.

Homogenization and multi-layer models (2025–2026)

The most recent work addresses materials with microstructure — composites and laminates — where effective properties must be computed from the fine-scale geometry.

Variational formulations in magnetostatics (2026)

Magnetic materials are commonly described in two ways. One formulation uses the magnetic induction as its basic unknown, whereas the other separates the material response, represented by the magnetization, from the magnetic field. Although both descriptions refer to the same physical system, their energy functionals have different mathematical structures. Understanding their relationship is important when deriving constitutive models, proving the existence of equilibria, or choosing variables for numerical computation.

In Krömer & Tomassetti (2026), On the single field formulation in magnetostatics (Mathematics and Mechanics of Solids, accepted for publication), we use convex duality to connect these formulations. We determine when their stationary points correspond and show how convexity, concavity, and relaxation change when one passes from one set of magnetic variables to the other. This provides a common variational framework for models that can otherwise appear unrelated and supplies a foundation for future work on coupled magnetoelastic equilibrium.

Summary and open questions

The progression in this research line follows a clear logic: from the thermodynamics of phase transitions in rigid magnets → to large-deformation rod and beam theories for soft MREs → to the inverse (design) problems of shape programming and form finding → to rigorous homogenization of composite microstructures and the variational foundations of magnetostatics. Open questions include the optimal 3D topology of magnetic inclusions for maximal actuation work, the coupling between growth and magnetization, and the extension of dimension-reduction results to shells.