elastic anisotropy of hcp metal crystals and polycrystals

IJRRAS 6 (4) ● March 2011 Tromans ● Elastic Anisotropy of Hcp Metal Crystals and Polycrystals
S 44 S
55 when referred to the transformed axes [Voigt 1928]. Furthermore, transformation of compliances to the
new axes must be conducted in the full tensor notation (see Eq. (5)) after which values may be converted to the
contracted matrix notation [Nye 1985]. Transformation is a tedious procedure aided by the resulting cylindrical
symmetry of the compliances with respect to the c-axis of the unit cell. Calculations of the transformed compliances
were first conducted by Voigt [1928, pg. 746-747] to yield Eqs. (13) and (14):
x 1
x
x3
z
y
N
x2
469
x 1'
x1 --
[2110]
[0001]
x3
o
(a) (b)
x3'
-
x2 [0110]
Figure 5. (a) Direction (θ degrees) of the normal N to the plane xyz with respect to X3 (b) Transformed orthogonal axes X 1 ,
X 2 and 3
X such that X3 is rotated by θ to coincide with N direction.
4
4
2 2
S S (sin )
S (cos )
( 2S
S )(cos )(sin
)
(13)
33
11
33
13
2
2 2
S (
G S S ) / 2 S ( S S 0.
5S
)(sin )
2(
S S 2S
S )(cos sin
)
(14)
44
55
44
11
12
44
Voigt’s [1928] original equations were written in terms of cos 2 θ and (1-cos 2 θ). In Eqs. (13) and (14) the original (1cos
2 θ) terms have been replaced by sin 2 θ. Note that when θ equals 0 o and 90 o 1
, Eq. (13) shows that ( S is
equivalent to (S33) -1 and (S11) -1 , respectively, consistent with Eq. (10). Similarly, in regard to Eq. (14), G
equivalent to S44 when θ = 0 o , and S G is equivalent to ( S44 2S11
2S12)
/ 2 when θ = 90 o , consistent with Eq. (12).
44
11
33
13
44
x 2'
33)
S is
1
1
Based on Eqs. (13) and (14), together with the relationships E ( S33)
and G ( S
G ) the angular variations of
Young’s modulus (E ) and the shear modulus (G )for all metals listed in Table 1 may be represented graphically, via
θ versus E and θ versus G diagrams . This is shown for the three metals Zn, Mg and Cd in Fig. 6 where the main
consideration determining the combination was a reasonable similarity in magnitude between the E moduli and G
moduli of each metal. It is readily evident that Zn and Cd are markedly anisotropic in behaviour, particularly Zn,
whereas Mg tends to be considerably less anisotropic. Furthermore, the cylindrical symmetry of the behaviours of E
and G with respect to the X3 axis (i.e. c-axis, [0001] direction) is evident in Fig 6, and evident in subsequent Figs. 7
and 8, via the mirror image of the angular data on either side of the X3 axis.
E (GPa)
100
50
0
Zn Zn
Cd
Mg
X3 N - + N
-90 -60 -30 0 30 60 90
(degrees)
Cd
Mg
G (GPa)
50
40
30
20
10
0
Zn Zn
Cd
Mg
X3 N - + N
-90 -60 -30 0 30 60 90
(degrees)
Figure 6. Angular variation of E and G for Zn, Cd and Mg
Cd
Mg