Striped periodic minimizers of a two-dimensional model for martensitic phase transitions

Abstract

In this paper we consider a simplified two-dimensional scalar model for the formation of mesoscopic domain patterns in martensitic shape-memory alloys at the interface between a region occupied by the parent (austenite) phase and a region occupied by the product (martensite) phase, which can occur in two variants (twins). The model, first proposed by Kohn and Mueller, is defined by the following functional: $$\mathcal{E}(u)=\beta ||u(0,\cdot )|{|}_{H^{1/2}([0,h])}^2+{\int }_0^{L}dx{\int }_0^{h}dy(|u_x{|}^2+\epsilon |u_{yy}|)$$ where $u:[0,L]\times [0,h]\to R$ is periodic in $y$ and $u_y=\pm 1$ almost everywhere. Conti proved that if $\beta \gtrsim \epsilon L/h^2$ then the minimal specific energy scales like $\sim \min \{(\epsilon \beta /L{)}^{1/2},(\epsilon /L{)}^{2/3}\}$, as $(\epsilon /L)\to 0$. In the regime $(\epsilon \beta /L{)}^{1/2}\ll (\epsilon /L{)}^{2/3}$, we improve Conti's results, by computing exactly the minimal energy and by proving that minimizers are periodic one-dimensional sawtooth functions.

29 pages, 3 figures

References in corpus (6)

• Ising models with long-range dipolar and short-range ferromagnetic interactions
• Crystalline ground states for classical particles
• Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions
• Striped phases in two dimensional dipole systems
• Periodic minimizers in 1D local mean field theory
• Modulated phases of a 1D sharp interface model in a magnetic field

Cited by in corpus (5)

• Exact periodic stripes for a minimizers of a local/non-local interaction functional in general dimension
• Periodic striped ground states in Ising models with competing interactions
• Realization of stripes and slabs in two and three dimensions
• Formation of stripes and slabs near the ferromagnetic transition
• Froth-like minimizers of a non local free energy functional with competing interactions