Hexagonal (hP)#

Stiffness Matrix#

\[\begin{split}\begin{pmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ & C_{11} & C_{13} & 0 & 0 & 0 \\ & & C_{33} & 0 & 0 & 0 \\ & & & C_{44} & 0 & 0 \\ & & & & C_{44} & 0 \\ & & & & & \frac{C_{12}-C_{12}}{2} \end{pmatrix}\end{split}\]

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

\([\bar 1 \bar 1 2 0]\)

\((0 0 0 1)\)

basal slip system

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

\([\bar 1 \bar 1 2 0]\)

\((1 \bar 1 0 0)\)

prismatic slip system

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

\([1 \bar 2 1 0]\)

\((1 0 \bar 1 1)\)

1st order pyramidal <a> slip system

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

\([\bar 1 \bar 1 2 3]\)

\((0 1 \bar 1 1)\)

1st order pyramidal <c+a> slip system

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

\([1 1 \bar 2 3]\)

\((\bar 1 \bar 1 2 2)\)

2nd order pyramidal <c+a> slip system

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

\([0 \bar 1 1 1]\)

\((0 1 \bar 1 2)\)

twin system

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

\([1 1 \bar 2 6]\)

\((\bar 1 \bar 1 2 1)\)

twin system

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

\([0 1 \bar 1 \bar 2]\)

\((0 1 \bar 1 1)\)

twin system

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

\([1 1 \bar 2 \bar 3]\)

\((1 1 \bar 2 2)\)

twin system

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.