Hexagonal (hP)#
Stiffness Matrix#
Atom Arrangement#
Figure 1: Hexagonal lattice structure. X, Y, and Z crystal frame axes are colored red, green, and blue, respectively.
Slip Systems#
index |
slip direction |
plane normal |
|---|---|---|
1 |
\([2 \bar 1 \bar 1 0]\) |
\((0 0 0 1)\) |
2 |
\([\bar 1 2 \bar 1 0]\) |
\((0 0 0 1)\) |
3 |
\([\bar 1 \bar 1 2 0]\) |
\((0 0 0 1)\) |
Figure 2: \(⟨1 1 \bar 2 0⟩\{0 0 0 1\}\) basal slip system
index |
slip direction |
plane normal |
|---|---|---|
4 |
\([2 \bar 1 \bar 1 0]\) |
\((0 \bar 1 \bar 1 0)\) |
5 |
\([\bar 1 2 \bar 1 0]\) |
\((\bar 1 0 0 1)\) |
6 |
\([\bar 1 \bar 1 2 0]\) |
\((1 \bar 1 0 0)\) |
Figure 3: \(⟨1 1 \bar 2 0⟩\{1 \bar 1 0 0\}\) prismatic slip system
index |
slip direction |
plane normal |
|---|---|---|
7 |
\([2 \bar 1 \bar 1 0]\) |
\((0 1 \bar 1 1)\) |
8 |
\([\bar 1 2 \bar 1 0]\) |
\((\bar 1 0 1 1)\) |
9 |
\([\bar 1 \bar 1 2 0]\) |
\((1 \bar 1 0 1)\) |
10 |
\([1 1 \bar 2 0]\) |
\((\bar 1 1 0 1)\) |
11 |
\([\bar 2 1 1 0]\) |
\((0 \bar 1 1 1)\) |
12 |
\([1 \bar 2 1 0]\) |
\((1 0 \bar 1 1)\) |
Figure 4: \(⟨1 1 \bar 2 0⟩\{1 0 \bar 1 1\}\) pyramidal \(\langle a \rangle\) slip system
index |
slip direction |
plane normal |
|---|---|---|
13 |
\([2 \bar 1 \bar 1 3]\) |
\((\bar 1 1 0 1)\) |
14 |
\([\bar 1 2 \bar 1 3]\) |
\((\bar 1 1 0 1)\) |
15 |
\([\bar 1 \bar 1 2 3]\) |
\((1 0 \bar 1 1)\) |
16 |
\([\bar 2 1 1 3]\) |
\((1 0 \bar 1 1)\) |
17 |
\([\bar 1 2 \bar 1 3]\) |
\((0 1 \bar 1 1)\) |
18 |
\([1 1 \bar 2 3]\) |
\((0 \bar 1 1 1)\) |
19 |
\([2 \bar 1 \bar 1 3]\) |
\((1 \bar 1 0 1)\) |
20 |
\([\bar 1 2 \bar 1 3]\) |
\((1 \bar 1 0 1)\) |
21 |
\([1 1 \bar 2 3]\) |
\((\bar 1 0 1 1)\) |
22 |
\([2 \bar 1 \bar 1 3]\) |
\((\bar 1 0 1 1)\) |
23 |
\([1 \bar 2 1 3]\) |
\((0 1 \bar 1 1)\) |
24 |
\([\bar 1 \bar 1 2 3]\) |
\((0 1 \bar 1 1)\) |
Figure 5: \(⟨1 1 \bar 2 3⟩\{1 0 \bar 1 1\}\) 1st order pyramidal \(\langle c + a \rangle\) slip system
index |
slip direction |
plane normal |
|---|---|---|
25 |
\([2 \bar 1 \bar 1 3]\) |
\((\bar 2 1 1 2)\) |
26 |
\([\bar 1 2 \bar 1 3]\) |
\((1 \bar 2 1 2)\) |
27 |
\([\bar 1 \bar 1 2 3]\) |
\((1 1 \bar 2 2)\) |
28 |
\([\bar 2 1 1 3]\) |
\((2 \bar 1 \bar 1 2)\) |
29 |
\([1 \bar 2 1 3]\) |
\((\bar 1 2 \bar 1 2)\) |
30 |
\([1 1 \bar 2 3]\) |
\((\bar 1 \bar 1 2 2)\) |
Figure 6: \(⟨1 1 \bar 2 3⟩\{1 1 \bar 2 2\}\) 2nd order pyramidal \(\langle c + a \rangle\) slip system
Twin Systems#
\(η_1\) |
\(K_1\) |
\(η_2\) |
\(K_2\) |
|---|---|---|---|
\(⟨\bar 1 0 1 1⟩\) |
\(\{1 0 \bar 1 2\}\) |
\(⟨1 0 \bar 1 1⟩\) |
\(\{1 0 \bar 1 \bar 2\}\) |
index |
slip direction |
plane normal |
|---|---|---|
1 |
\([1 \bar 1 0 1]\) |
\((\bar 1 1 0 2)\) |
2 |
\([\bar 1 0 1 1]\) |
\((1 0 \bar 1 2)\) |
3 |
\([0 1 \bar 1 1]\) |
\((0 \bar 1 1 2)\) |
4 |
\([\bar 1 1 0 1]\) |
\((1 \bar 1 0 2)\) |
5 |
\([1 0 \bar 1 1]\) |
\((\bar 1 0 1 2)\) |
6 |
\([0 \bar 1 1 1]\) |
\((0 1 \bar 1 2)\) |
Figure 7: \(⟨\bar 1 0 1 1⟩ \{1 0 \bar 1 2\}\) T1 tensile twinning in Co, Mg, Zr, Ti, and Be; compressive twinning in Cd and Zn.
\(η_1\) |
\(K_1\) |
\(η_2\) |
\(K_2\) |
|---|---|---|---|
\(⟨\bar 1 \bar 1 2 6⟩\) |
\(\{1 1 \bar 2 1\}\) |
\(⟨1 1 2 0⟩\) |
\(\{0 0 0 2\}\) |
index |
slip direction |
plane normal |
|---|---|---|
7 |
\([2 \bar 1 \bar 1 6]\) |
\((\bar 2 1 1 1)\) |
8 |
\([\bar 1 2 \bar 1 6]\) |
\((1 1 \bar 2 1)\) |
9 |
\([\bar 1 \bar 1 2 6]\) |
\((2 \bar 1 \bar 1 1)\) |
10 |
\([\bar 2 1 1 6]\) |
\((\bar 1 2 \bar 1 1)\) |
11 |
\([1 \bar 2 1 6]\) |
\((\bar 1 0 1 2)\) |
12 |
\([1 1 \bar 2 6]\) |
\((\bar 1 \bar 1 2 1)\) |
Figure 8: \(⟨\bar 1 \bar 1 2 6⟩ \{1 1 \bar 2 1\}\) T2 tensile twinning in Co, Re, and Zr.
\(η_1\) |
\(K_1\) |
\(η_2\) |
\(K_2\) |
|---|---|---|---|
\(⟨1 0 \bar 1 \bar 2⟩\) |
\(\{1 0 \bar 1 1\}\) |
\(⟨3 0 \bar 3 2⟩\) |
\(\{1 0 \bar 1 \bar 3\}\) |
index |
slip direction |
plane normal |
|---|---|---|
13 |
\([\bar 1 1 0 \bar 2]\) |
\((\bar 1 1 0 1)\) |
14 |
\([1 0 \bar 1 \bar 2]\) |
\((1 0 \bar 1 1)\) |
15 |
\([0 \bar 1 1 \bar 2]\) |
\((0 \bar 1 1 1)\) |
16 |
\([1 \bar 1 0 \bar 2]\) |
\((1 \bar 1 0 1)\) |
17 |
\([\bar 1 0 1 \bar 2]\) |
\((\bar 1 0 1 1)\) |
18 |
\([0 1 \bar 1 \bar 2]\) |
\((0 1 \bar 1 1)\) |
Figure 9: \(⟨1 0 \bar 1 \bar 2⟩ \{1 0 \bar 1 1\}\) C1 compressive twinning in Mg.
\(η_1\) |
\(K_1\) |
\(η_2\) |
\(K_2\) |
|---|---|---|---|
\(⟨1 1 \bar 2 \bar 3⟩\) |
\(\{1 1 \bar 2 2\}\) |
\(⟨2 2 \bar 4 3⟩\) |
\(\{1 1 \bar 2 \bar 4\}\) |
index |
slip direction |
plane normal |
|---|---|---|
19 |
\([2 \bar 1 \bar 1 \bar 3]\) |
\((2 \bar 1 \bar 1 2)\) |
20 |
\([\bar 1 2 \bar 1 \bar 3]\) |
\((\bar 1 2 \bar 1 2)\) |
21 |
\([\bar 1 \bar 1 2 \bar 3]\) |
\((\bar 1 \bar 1 2 2)\) |
22 |
\([\bar 2 1 1 \bar 3]\) |
\((\bar 2 1 1 2)\) |
23 |
\([1 \bar 2 1 \bar 3]\) |
\((1 \bar 2 1 2)\) |
24 |
\([1 1 \bar 2 \bar 3]\) |
\((1 1 \bar 2 2)\) |
Figure 10: \(⟨1 1 \bar 2 \bar 3⟩ \{1 1 \bar 2 2\}\) C2 compressive twinning in Ti and Zr.
Interaction Matrices#
Slip-Slip#
index |
label |
description |
|---|---|---|
1 |
S1 |
basal self-interaction |
2 |
1 |
basal/basal coplanar |
3 |
3 |
basal/prismatic collinear |
4 |
4 |
basal/prismatic non-collinear |
5 |
S2 |
prismatic self-interaction |
6 |
2 |
prismatic/prismatic |
7 |
5 |
prismatic/basal collinear |
8 |
6 |
prismatic/basal non-collinear |
9 |
\(-\) |
basal/pyramidal \(\langle a \rangle\) non-collinear |
10 |
\(-\) |
basal/pyramidal \(\langle a \rangle\) collinear |
11 |
\(-\) |
prismatic/pyramidal \(\langle a \rangle\) non-collinear |
12 |
\(-\) |
prismatic/pyramidal \(\langle a \rangle\) collinear |
13 |
\(-\) |
pyramidal \(\langle a \rangle\) self-interaction |
14 |
\(-\) |
pyramidal \(\langle a \rangle\) non-collinear |
15 |
\(-\) |
pyramidal \(\langle a \rangle\) collinear |
16 |
\(-\) |
pyramidal \(\langle a \rangle\)/prismatic non-collinear |
17 |
\(-\) |
pyramidal \(\langle a \rangle\)/prismatic collinear |
18 |
\(-\) |
pyramidal \(\langle a \rangle\)/basal non-collinear |
19 |
\(-\) |
pyramidal \(\langle a \rangle\)/basal collinear |
20 |
\(-\) |
basal/1. order pyramidal \(\langle c+a \rangle\) semi-collinear |
21 |
\(-\) |
basal/1. order pyramidal \(\langle c+a \rangle\) |
22 |
\(-\) |
basal/1. order pyramidal \(\langle c+a \rangle\) |
23 |
\(-\) |
prismatic/1. order pyramidal \(\langle c+a \rangle\) semi-collinear |
24 |
\(-\) |
prismatic/1. order pyramidal \(\langle c+a \rangle\) |
25 |
\(-\) |
prismatic/1. order pyramidal \(\langle c+a \rangle\) semi-coplanar? |
26 |
\(-\) |
pyramidal \(\langle a \rangle\)/1. order pyramidal \(\langle c+a \rangle\) coplanar |
27 |
\(-\) |
pyramidal \(\langle a \rangle\)/1. order pyramidal \(\langle c+a \rangle\) |
28 |
\(-\) |
pyramidal \(\langle a \rangle\)/1. order pyramidal \(\langle c+a \rangle\) semi-collinear |
29 |
\(-\) |
pyramidal \(\langle a \rangle\)/1. order pyramidal \(\langle c+a \rangle\) semi-coplanar |
30 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\) self-interaction |
31 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\) coplanar |
32 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\) |
33 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\) |
34 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\) semi-coplanar |
35 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\) semi-coplanar |
36 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\) collinear |
37 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/pyramidal \(\langle a \rangle\) coplanar |
38 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/pyramidal \(\langle a \rangle\) semi-collinear |
39 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/pyramidal \(\langle a \rangle\) |
40 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/pyramidal \(\langle a \rangle\) semi-coplanar |
41 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/prismatic semi-collinear |
42 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/prismatic semi-coplanar |
43 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/prismatic |
44 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/basal semi-collinear |
45 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/basal |
46 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/basal |
47 |
8 |
basal/2. order pyramidal \(\langle c+a \rangle\) non-collinear |
48 |
7 |
basal/2. order pyramidal \(\langle c+a \rangle\) semi-collinear |
49 |
10 |
prismatic/2. order pyramidal \(\langle c+a \rangle\) |
50 |
9 |
prismatic/2. order pyramidal \(\langle c+a \rangle\) semi-collinear |
51 |
\(-\) |
pyramidal \(\langle a \rangle\)/2. order pyramidal \(\langle c+a \rangle\) |
52 |
\(-\) |
pyramidal \(\langle a \rangle\)/2. order pyramidal \(\langle c+a \rangle\) semi collinear |
53 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/2. order pyramidal \(\langle c+a \rangle\) |
54 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/2. order pyramidal \(\langle c+a \rangle\) |
55 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/2. order pyramidal \(\langle c+a \rangle\) |
56 |
\(-\) |
1. order pyramidal \(\langle c+a \rangle\)/2. order pyramidal \(\langle c+a \rangle\) collinear |
57 |
S3 |
2. order pyramidal \(\langle c+a \rangle\) self-interaction |
58 |
16 |
2. order pyramidal \(\langle c+a \rangle\) non-collinear |
59 |
15 |
2. order pyramidal \(\langle c+a \rangle\) semi-collinear |
60 |
\(-\) |
2. order pyramidal \(\langle c+a \rangle\)/1. order pyramidal \(\langle c+a \rangle\) |
61 |
\(-\) |
2. order pyramidal \(\langle c+a \rangle\)/1. order pyramidal \(\langle c+a \rangle\) collinear |
62 |
\(-\) |
2. order pyramidal \(\langle c+a \rangle\)/1. order pyramidal \(\langle c+a \rangle\) |
63 |
\(-\) |
2. order pyramidal \(\langle c+a \rangle\)/1. order pyramidal \(\langle c+a \rangle\) |
64 |
\(-\) |
2. order pyramidal \(\langle c+a \rangle\)/pyramidal \(\langle a \rangle\) non-collinear |
65 |
\(-\) |
2. order pyramidal \(\langle c+a \rangle\)/pyramidal \(\langle a \rangle\) semi-collinear |
66 |
14 |
2. order pyramidal \(\langle c+a \rangle\)/prismatic non-collinear |
67 |
13 |
2. order pyramidal \(\langle c+a \rangle\)/prismatic semi-collinear |
68 |
12 |
2. order pyramidal \(\langle c+a \rangle\)/basal non-collinear |
69 |
11 |
2. order pyramidal \(\langle c+a \rangle\)/basal semi-collinear |
N. Bertin, C.N. Tomé, I.J. Beyerlein, M.R. Barnett, and L. Capolungo. On the strength of dislocation interactions and their effect on latent hardening in pure Magnesium International Journal of Plasticity, 62:72-92, 2014. doi:10.1016/j.ijplas.2014.06.010.
B. Devincre. Dislocation dynamics simulations of slip systems interactions and forest strengthening in ice single crystal Philosophical Magazine A, 93(1-3):1-12, 2013. doi:10.1080/14786435.2012.699689.