Cosserat micropolar and couple-stress elasticity models of flexomagnetism at finite deformations

TL;DR

This work develops geometrically nonlinear flexomagnetic models based on Cosserat micropolar and its couple-stress descendants, coupling micro-dislocation tensors to magnetization through a Lifshitz invariant. By formulating both scalar and vector potential magnetostatics and employing a Legendre transform to unify magnetic variables, the authors derive fully coupled, finite-deformation governing equations and demonstrate numerical feasibility on a chromia nano-beam. Key findings show that flexomagnetic effects arise from curvature (micro-rotation) rather than nonuniform extension, with the coupling strength controlled by a small set of material parameters and sensitive to symmetry (centrosymmetric vs cubic). The work provides a coherent variational framework and numerical strategy for calibrating and applying finite-deformation flexomagnetism in nanoscale magneto-mechanical devices, while highlighting the need for experimental or first-principles parameter determination.

Abstract

We propose geometrically nonlinear (finite) continuum models of flexomagnetism based on the Cosserat micropolar and its descendent couple-stress theory. These models introduce the magneto-mechanical interaction by coupling the micro-dislocation tensor of the micropolar model with the magnetisation vector using a Lifshitz invariant. In contrast to conventional formulations that couple strain-gradients to the magnetisation using fourth-order tensors, our approach relies on third-order tensor couplings by virtue of the micro-dislocation being a second-order tensor. Consequently, the models permit centrosymmetric materials with a single new flexomagnetic constant, and more generally allow cubic-symmetric materials with two such constants. We postulate the flexomagnetic action-functionals and derive the corresponding governing equations using both scalar and vectorial magnetic potential formulations, and present numerical results for a nano-beam geometry, confirming the physical plausibility and computational feasibility of the models.

Figures (10)

• Figure 1: Mapping of the reference configuration $V \subset \mathbb{R}^3$ onto the current configuration $V_\varphi \subset \mathbb{R}^3$ by the deformation tensor $\bm{F}: V \to \mathrm{GL}^+(3)$. The mapping can be decomposed into the initial application of the Biot-type stretch tensor $\overline{\bm{U}}:V \to \mathrm{GL}^+(3)$, followed by an independent rotation of the material points $\overline{\bm{R}}:V \to \mathrm{SO}(3)$. Note that in general, the tangential curves of the various configurations do not agree with the independent orientation of each material point. The correspondence between the two is predominantly governed by the Cosserat couple modulus $\mu_{\mathrm{c}}$, which can be roughly understood as a rotational spring.
• Figure 2: A net-magnetisation induced by a non-uniform change in the orientation of magnetic dipoles over a non-negligible region of the domain. The change in orientation is governed by $\overline{\bm{R}}:V \to \mathrm{SO}(3)$, while its non-uniformity and impact with respect to the size of the domain is indicated by the curvature energy $\Psi_\mathrm{curv}(\overline{\bm{\mathfrak{A}}})$.
• Figure 3: (a) Cantilever nano-beam with Dirichlet and Neumann boundary surfaces. (b) Mesh of $20$ hexahedral finite elements with polynomial orders $p \geq 3$, as per field.
• Figure 4: (a) Maximal bending deformation of the beam depending on $\mu_{\mathrm{c}}$ and $L_\mathrm{c}$ . Beam deformation with $\mu_{\mathrm{c}}/\mu_\mathrm{e} = 1$ for $L_\mathrm{c} = 1$ (b) and $L_\mathrm{c} = 10^2$ (c).
• Figure 5: (a) Maximal torsion deformation of the beam depending on $\mu_{\mathrm{c}}$ and $L_\mathrm{c}$ . Beam deformation with $\mu_{\mathrm{c}}/\mu_\mathrm{e} = 1$ for $L_\mathrm{c} = 1$ (b) and $L_\mathrm{c} = 10^2$ (c).

Theorems & Definitions (2)

• Remark 3.1
• Remark 4.1