Micromechanical stress calculations for {10-12} extension twin embryos in single-crystal Magnesium (MEAM potential stiffness).
Analytical 2D Ellipse (AR 4:1)
-127.21 MPa
Pure twinning: -586.38 MPa • Shear relief: +459.17 MPa
Analytical 2D Circle (1:1)
-490.50 MPa
Pure twinning: -932.21 MPa • Shear relief: +441.71 MPa
FEM 105nm Hexagonal Twin
-171.14 MPa
Faceted (10-12) boundaries in 2.0 μm box (Isolated limit: -170.04 MPa)
Analytical 3D Sphere (1:1:1)
-1016.81 MPa
Pure twinning: -1439.74 MPa • FEM 3D model: -979.79 MPa
Finite element parametric twin size scaling (12 nm to 520 nm in 2.0 μm box) compared against exact infinite-medium analytical Eshelby limits (dashed lines).
Exact inhomogeneous inclusion solution in anisotropic Magnesium with twinning eigenstrain γ* = 0.129 (ε*xy = 0.0645).
| Geometry / Inclusion Shape | Aspect Ratio (ax : ay : az) | Pure Twinning (γappl = 0) [Isotropic] | Pure Twinning (γappl = 0) [Anisotropic] | With 2% Applied Shear (γappl = 0.02) | Forward Shear Relief (Δσxy) |
|---|---|---|---|---|---|
| 2D Ellipse (1/4) | 1 : 0.25 (Plane Strain) | -553.98 | -586.38 | -127.21 | +459.17 |
| 3D Elliptical Plate | 1 : 0.25 : ∞ | -553.85 | -586.20 | -127.02 | +459.18 |
| 3D Ellipsoid (1/4) | 1 : 0.25 : 1 | -754.24 | -815.10 | -366.62 | +448.48 |
| 2D Circle | 1 : 1 (Plane Strain) | -865.60 | -932.21 | -490.50 | +441.71 |
| 3D Cylinder | 1 : 1 : ∞ | -865.76 | -932.34 | -490.62 | +441.72 |
| 3D Sphere | 1 : 1 : 1 | -1292.63 | -1439.74 | -1016.81 | +422.93 |
Comparing infinite-domain analytical Eshelby theory with finite-domain periodic FEM simulations.
| Model Geometry | Analytical Eshelby τ [MPa] | FEM Benchmark (105nm in 2.0 μm Box) | FEM Asymptotic Limit (Ltwin/Lbox → 0) | Agreement |
|---|---|---|---|---|
| Pure Ellipse (AR 4:1) | -127.21 | -122.86 | -121.84 | ~3.5% (Image stress effect) |
| Circular Twin (1:1) | -490.50 | -476.56 | -472.52 | ~3.0% (Image stress effect) |
| 3D Sphere (1:1:1) | -1016.81 | -979.79 | — | ~3.6% |
| Hexagonal Faceted Twin | — | -171.14 | -170.04 | Faceted continuum geometry |
Convergence evaluation comparing baseline twin boundary element sizing against 2× coarser boundary mesh across all 9 hexagonal twin sizes in 2.0 μm box.
| Twin Size (Ltw) | Baseline Boundary Sizing | Baseline Sxy [MPa] | 2× Boundary Sizing | 2× Mesh Sxy [MPa] | ΔSxy [MPa] | Relative Diff |
|---|---|---|---|---|---|---|
| 12 nm | 0.0571 nm | -170.0425 | 0.1142 nm | -170.0582 | -0.0156 | +0.009% |
| 25 nm | 0.1190 nm | -170.0848 | 0.2380 nm | -170.0964 | -0.0116 | +0.007% |
| 36 nm | 0.1710 nm | -170.1584 | 0.3420 nm | -170.1778 | -0.0194 | +0.011% |
| 50 nm | 0.2380 nm | -170.2816 | 0.4760 nm | -170.3019 | -0.0203 | +0.012% |
| 75 nm | 0.3570 nm | -170.5965 | 0.7140 nm | -170.6083 | -0.0118 | +0.007% |
| 105 nm | 0.5000 nm | -171.1404 | 1.0000 nm | -171.1620 | -0.0216 | +0.013% |
| 210 nm | 0.5000 nm | -174.4351 | 1.0000 nm | -174.4399 | -0.0048 | +0.003% |
| 340 nm | 0.5000 nm | -181.3553 | 1.0000 nm | -181.3578 | -0.0025 | +0.001% |
| 520 nm | 0.5000 nm | -195.3888 | 1.0000 nm | -195.3896 | -0.0008 | +0.000% |