Magnetoelasticity and Soft Actuators
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 and rigorous dimension reduction.
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:
- Podio-Guidugli & Tomassetti (2002) derived the steady-state velocity of a flat domain wall subject to an applied field, showing how viscous dissipation and exchange energy control the motion.
- Podio-Guidugli & Tomassetti (2004) extended this to curvature-driven motion, establishing an existence result for the full evolution problem.
- Podio-Guidugli & Tomassetti (2006) studied magnetization switching under non-standard dissipation, relevant to the design of magnetic memory devices.
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.
- Roubíček & Tomassetti (2010) developed a thermodynamically consistent model for SMAs under electric current, establishing well-posedness for the coupled PDE system.
- Podio-Guidugli, Roubíček & Tomassetti (2010) provided a thermodynamic theory of the ferro/paramagnetic (Curie) transition, deriving it from a Landau-type free energy and proving existence of solutions.
- Roubíček & Tomassetti (2011) studied ferromagnets with eddy currents and pinning effects, establishing an energy-dissipation balance and existence theory.
- Roubíček & Tomassetti (2013) treated phase transformations in electrically conductive ferromagnetic SMAs at large strains, a technically demanding problem solved by combining techniques from plasticity and micromagnetics.
Magneto-rheological elastomers: large-deformation rod and beam theories (2018–2022)
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.
- Ciambella, Favata & Tomassetti (2018) derived the first geometrically exact theory for a fibre-reinforced magneto-elastic rod: the fibre architecture couples magnetic susceptibility to the rod’s axis direction, producing a preferred bending plane that can be steered by the field. This model captured the “huge magnetostriction” effect observed experimentally.
- Building on this rod theory, Ciambella & Tomassetti (2020) turned the problem around: rather than analyzing the response to a given field, they asked which field distribution produces a prescribed deformed shape (the form-finding problem). This is the natural question for actuator design, and the paper provided a systematic variational answer.
- Durastanti, Giacomelli & Tomassetti (2021) addressed the shape-programming problem for a magnetic elastica (an inextensible rod with bending stiffness): given a target shape, find the remanent magnetization to imprint during fabrication so that the rod assumes that shape under a uniform field. The paper proved existence and uniqueness of the optimal magnetization profile and characterized it via a variational inequality.
- Lembo & Tomassetti (2022) revisited the equilibrium of Kirchhoff rods subject to distributed magnetic couples, clarifying the role of the couple-stress in the rod’s balance laws and deriving exact solutions for helical equilibria.
- Ciambella, Kružík & Tomassetti (2022) performed a rigorous dimension reduction from 3D magnetoelasticity to a 1D nanorod model using Γ-convergence, establishing that the limiting rod theory inherits a well-defined magnetic energy that could not have been postulated from purely structural considerations.
Homogenization and multi-layer models (2025)
The most recent work addresses materials with microstructure — composites and laminates — where effective properties must be computed from the fine-scale geometry.
- Grande, Krömer, Kružík & Tomassetti (2026) — Homogenization of magnetoelastic materials with rigid magnetic inclusions at small strains (Journal of Nonlinear Science, 36(1), 4). This paper derives effective equations for a magnetoelastic composite with periodic rigid magnetic inclusions via two-scale homogenization at small strains, showing that the macroscopic coupling tensor depends on the inclusion geometry in a non-trivial way.
- Ruggieri & Tomassetti (2025) studied magneto-viscoelastic laminates, extending the theory to account for time-dependent relaxation effects, with implications for actuators subject to cyclic loading.
- Rubin & Tomassetti (2025) developed an Eulerian Cosserat shell theory that incorporates surface growth, unifying the magnetoelastic shell framework with the accretion mechanics developed in a parallel research thread (see Surface Growth).
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. 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.