A geometrically nonlinear theory of flexomagnetism — magnetism induced by deformation alone, with no coils, no current, and no applied field.
Flexomagnetism is a magneto-mechanical effect: a non-uniform deformation — a strain gradient, not just strain — generates a magnetic response on its own.
Unlike magnetostriction, which only runs one way — apply a field, get a shape change — flexomagnetism is a two-way coupling. Bend the material and it magnetizes; the magnetic state feeds back into the mechanics. Because it needs no external field, current, or time-varying voltage, it's attracted interest for energy harvesting, electronics, and magneto-mechanical devices at small scales, where strain gradients get large and the effect stops being negligible.
Conventional continuum treatments couple the strain gradient directly to the magnetization vector. Since a strain gradient is already a third-order object, the coupling tensor that connects it to magnetization comes out fourth-order: more independent material constants to identify, and a heavier structure to carry through a geometrically nonlinear, finite-deformation formulation.
The new models are built on the Cosserat micropolar continuum instead of classical elasticity. Every material point carries its own micro-rotation, independent of the surrounding deformation. The natural gradient-type quantity here is the micro-dislocation tensor — already second-order, one order lighter than a classical strain gradient.
Coupling that second-order tensor to the magnetization vector through a Lifshitz invariant needs only a third-order coupling tensor — one order down from the conventional formulation, with far fewer constants to calibrate.
The paper works out both the full micropolar model, where micro-rotation is an independent field, and its couple-stress descendant, where that rotation is slaved to the deformation. Each is posed with both a scalar and a vector magnetic potential formulation — giving four internally consistent variants of one underlying theory.
Rotation is a free field at every material point, coupled to the deformation through the micro-dislocation tensor.
The descendant theory: micro-rotation follows the macroscopic deformation gradient directly.
Governing equations derived from a scalar magnetic potential formulation.
The same physics re-derived from a vector magnetic potential formulation.
The models are exercised on a nano-beam geometry, where strain gradients are large enough for the effect to matter. The numerical results confirm both the physical plausibility of the coupling and the computational feasibility of solving the resulting finite-deformation problem.