The paper deals with the dimensional reduction from 2D to 1D in magnetoelastic interactions. We adopt a simplified, but nontrivial model described by the Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We identify the limit problem by using the so-called energy method.

$\Gamma$-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness $\epsilon$ approaches zero of a ferromagnetic thin structure ${\Omega }_{\epsilon }=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {ℝ}^2$, whose energy is given by$${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\mathrm{d}x$$subject to$$\text{div}(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }})=0\text{on}{ℝ}^3,$$and to the constraint$$|\overline{m}|=1\text{on}{\Omega }_{\epsilon },$$where $W$ is any continuous function satisfying $p$-growth assumptions with $p\>1$. Partial results are also obtained in the case $p=1$, under an additional assumption on $W$.

Γ-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure ${\Omega }_{\epsilon }=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {ℝ}^2$, whose energy is given by $${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\mathrm{d}x$$ subject to $$\text{div}(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }})=0\text{on}{ℝ}^3,$$ and to the constraint $$|\overline{m}|=1\text{on}{\Omega }_{\epsilon },$$ where W is any continuous function satisfying p-growth assumptions with p> 1. Partial results are also obtained in the case p=1, under an additional assumption on W.