Lattice reduction in the Poincaré disk

Drag the tips of the current basis vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ to deform the lattice. Collect them as the columns of $\mathbf{A}=\left({\mathbf{a}}_1{\mathbf{a}}_2\right)$. The reference matrix is ${\mathbf{A}}^0=\left({\mathbf{a}}_1^0{\mathbf{a}}_2^0\right)$. The deformation gradient $\mathbf{F}$ relates the two bases through $\mathbf{A}=\mathbf{F}{\mathbf{A}}^0$.

Here $\mathbf{C}={\mathbf{F}}^{\mathrm{T}}\mathbf{F}$ is the right Cauchy–Green strain tensor. The square reference is ${{\mathbf{A}}^0}_{\square }=\mathbf{I}$; the triangular reference uses a unit-area equilateral basis [3]. Changing the reference preserves $\mathbf{F}$ and recenters the disk. Reset lattice sets $\mathbf{F}=\mathbf{I}$. Dragging a disk marker changes lattice shape while preserving the rotation of $\mathbf{F}$, signed cell area, and handedness.

Heatmap vector chooses which basis vector varies across the lattice background while the other stays fixed. Dragging a vector selects it too. The disk heatmap shows unit-area metrics. The $\beta$ slider controls the Conti potential independently of the reference, with snapping at −0.25 for square symmetry and 4 for triangular symmetry.

The Potential menu offers the Conti–Zanzotto polynomial [1] and the quadratic potential of Perchikov and Truskinovsky [2].

Plastic reduction follows the elastic-domain construction in [2]. Reduction steps shows discrete integer basis changes as dashed arrows with a marker at each step. These indicate the order of the changes rather than a continuous deformation. Local minima adds white crosses at stable wells within three unit-shear transitions; lattice-space crosses mark minima of the fixed-vector energy slice.

In Transformations, choose Simple shear, Pure shear, or Rotation. Real-space transformations multiply on the left, ${\mathbf{F}}^{\prime }=𝒮\mathbf{F}$; material transformations multiply on the right, ${\mathbf{F}}^{\prime }=\mathbf{F}\mathbf{M}$. Shears use $\gamma$; rotations use $\mathbf{R}\left(\theta \right)$. The operator matrix shows the last transformation applied. The equations also show how the right Cauchy–Green tensor changes; the numeric matrices show the current $\mathbf{F}$.

Arrow keys work anywhere on the page. Hold them for smooth, accelerating motion, or use Shift + arrows for unit shears. Left/Right and Down/Up follow the selected transformation pair. Pure shear has two editable directions, initially 0° and 45°. In Rotation mode, Left/Right rotate, Shift + Left/Right applies 5° steps, and Down/Up uniformly shrinks or grows both vectors. Shift + Down/Up divides or multiplies their lengths by 1.05. Uniform scaling has the same effect for left and right multiplication. The transformation buttons use Step $\gamma$ or Step $\theta$. The Rotation of $\mathbf{F}$ slider snaps near multiples of 30° and 45°, including 0°, 60°, and 90°.

Drag a plot’s background to pan, and scroll or pinch to zoom. Each ◐ button toggles that plot’s background; the Background menu selects the field. Use +/− or 0 to zoom or reset the focused plot, and R to reset the lattice. Resize views with the dividers or their lower-right corners.

Present opens a minimal fullscreen view; Escape returns to the controls. Visual settings has separate normal and presentation profiles, including background color and basis labels. Animation uses the latest settings of its selected profile throughout the preview and export; keyframes do not store visual profiles. Settings last for this session unless you choose to save them.

References

1. S. Conti and G. Zanzotto. “A variational model for reconstructive phase transformations in crystals, and their relation to dislocations and plasticity.” Archive for Rational Mechanics and Analysis 173(1), 69–88 (2004).
2. N. Perchikov and L. Truskinovsky. “Quantized plastic deformation.” Journal of the Mechanics and Physics of Solids 190, 105704 (2024).
3. E. Lundheim. Mesoscopic Tensorial Modelling of Crystal Plasticity: Mesh Reconnection, Elastoplastic Decomposition and Plastic Avalanches. Doctoral manuscript (2026).