A Kirchhoff-like theory for hard magnetic rods under geometrically nonlinear deformation in three dimensions

Tomohiko G. Sano, Matteo Pezzulla, Pedro M. Reis

Research output: Contribution to journalArticlepeer-review

40 Citations (Scopus)


Magneto-rheological elastomers (MREs) are functional materials that can be actuated by applying an external magnetic field. MREs comprise a composite of hard magnetic particles dispersed into a nonmagnetic elastomeric (soft) matrix. Hard MREs have been receiving particular attention because the programmed magnetization remains unchanged upon actuation. Motivated by a new realm of applications, there have been significant theoretical developments in the continuum (three-dimensional) description of hard MREs. In this paper, we derive an effective theory for MRE rods (slender, mono-dimensional structures) under geometrically nonlinear deformation in three dimensions. Our theory is based on reducing the three-dimensional magneto-elastic energy functional for the hard MREs into an one-dimensional Kirchhoff-like description (centerline-based). Restricting the theory to two dimensions, we reproduce previous works on planar deformations. For further validation in the general case of three-dimensional deformation, we perform precision experiments with both naturally straight and curved rods under either constant or constant-gradient magnetic fields. Our theoretical predictions are in excellent agreement with both discrete simulations and precision-model experiments. Finally, we discuss some limitations of our framework, as highlighted by the experiments, where long-range dipole–dipole interactions, which are neglected in the theory, can play a role.

Original languageEnglish
Article number104739
JournalJournal of the Mechanics and Physics of Solids
Publication statusPublished - 2022 Mar
Externally publishedYes


  • Buckling
  • Kirchhoff rod theory
  • Magnetorheological elastomer
  • Slender structures

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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