Anisotropic continuum simulations of Magnesium twin embryos: Pure Elliptical, Hexagonal Faceted, Circular, and 3D Spherical Twin Models.
Infinite Matrix Limit τ (Ellipse)
-121.84 MPa
Pure Ellipse (AR 4:1) asymptotic shear stress as Ltwin / Lbox → 0
Infinite Matrix Limit τ (Circle)
-472.52 MPa
Circular twin asymptotic shear stress as Dtwin / Lbox → 0
Geometric Scaling Invariance
< 0.01% Δτ
Invariant τ across 2.0 μm, 4.0 μm, and 10.0 μm boxes at constant size ratio
3D 105nm Spherical Twin τ
-979.79 MPa
3D spherical inclusion under 2.0% applied shear strain
Volume-averaged shear stress τ [MPa] plotted across geometries (12 nm to 520 nm).
Boundary decay of τ across 2.0 μm, 4.0 μm, and 10.0 μm boxes.
Comparison of internal shear stress τ [MPa] across twin geometries under 2% applied shear (γ = 0.02, utop = 0.04 μm).
| Twin Size | Hexagonal Faceted Twin τ [MPa] | Pure Elliptical Twin (AR 4:1) τ [MPa] | Circular Twin τ [MPa] |
|---|---|---|---|
| 12 nm | -170.0425 | -121.8586 | -472.5684 |
| 25 nm | -170.0848 | -121.9028 | -472.7451 |
| 36 nm | -170.1584 | -121.9644 | -472.9916 |
| 50 nm | -170.2816 | -122.0749 | -473.4338 |
| 75 nm | -170.5965 | -122.3614 | -474.5813 |
| 105 nm | -171.1404 | -122.8552 | -476.5628 |
| 210 nm | -174.4351 | -125.8468 | -488.6676 |
| 340 nm | -181.3553 | -132.1235 | -514.6526 |
| 520 nm | -195.3888 | -144.8637 | -570.1479 |
Applied shear displacement scales proportionally with domain box size (utop = γ × Lbox).
| Twin Size | Pure Ellipse τ (2.0 μm Box) | Pure Ellipse τ (4.0 μm Box) | Pure Ellipse τ (10.0 μm Box) | Circular Twin τ (2.0 μm Box) | Circular Twin τ (4.0 μm Box) | Circular Twin τ (10.0 μm Box) |
|---|---|---|---|---|---|---|
| 12 nm | -121.8586 | -121.8487 | — | -472.5684 | -472.5287 | — |
| 25 nm | -121.9028 | -121.8597 | — | -472.7451 | -472.5729 | — |
| 36 nm | -121.9644 | -121.8751 | — | -472.9916 | -472.6345 | — |
| 50 nm | -122.0749 | -121.9027 | — | -473.4338 | -472.7451 | — |
| 75 nm | -122.3614 | -121.9745 | — | -474.5813 | -473.0321 | — |
| 105 nm | -122.8552 | -122.0984 | — | -476.5628 | -473.5279 | — |
| 210 nm | -125.8468 | -122.8552 | — | -488.6676 | -476.5628 | — |
| 340 nm | -132.1235 | -124.4790 | — | -514.6526 | -483.1116 | — |
| 520 nm | -144.8637 | -127.9382 | -122.8361 | -570.1479 | -497.2350 | -476.4862 |
• Geometric Invariance Highlight:
Holding the normalized ratio Ltwin / Lbox = 0.0525 constant yields identical shear stress across all scales:
• 105nm in 2.0 μm Box: τellipse = -122.8552 MPa, τcircle = -476.5628 MPa
• 210nm in 4.0 μm Box: τellipse = -122.8552 MPa, τcircle = -476.5628 MPa
• 520nm in 10.0 μm Box: τellipse = -122.8361 MPa, τcircle = -476.4862 MPa
High-order 3D solid elements (SOLID186/SOLID187), 3D periodic boundary conditions, 2% applied shear.
Volume-Averaged Shear Stress τ
-979.7880 MPa
Volume Conservation Accuracy
99.998%
Simulated V = 6.0612e-4 μm3