elastic anisotropy of hcp metal crystals and polycrystals

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IJRRAS 6 (4) ● March 2011 Tromans ● Elastic Anisotropy of Hcp Metal Crystals and Polycrystals E (GPa) 800 700 600 500 400 300 200 100 0 Os Os Ru Re Re Be Be -90 -60 -30 0 30 60 90 (degrees) Ru Co Co X3 N - + N 471 G (GPa) 300 250 200 150 100 50 0 Os Os Ru Re Be Be -90 -60 -30 0 30 60 90 (degrees) Re Ru Co X3 Co N - + N Figure 8. Angular variation of E and G for Os, Ru, Re, Be and Co The presence and precise position of an intermediate maximum or minimum E at 0 < θ < 90 degrees in most of the graphical plots in Figs 6 to 8, due to a minimum or maximum in S 33 , was ascertained by differentiating Eq. (13) and placing the first differential equal to zero: S / 4S 3 sin cos 4S 3 cos sin 2( 2S S 33 11 o o with solutions 0 , 90 , and tan [( S 33 44 13 2S 13 44 2S 3 3 )(sin cos cos sin ) 0 Similar procedures were applied to calculate the position of intermediate maxima/minima for G at 0< θ
IJRRAS 6 (4) ● March 2011 Tromans ● Elastic Anisotropy of Hcp Metal Crystals and Polycrystals Table 5. Calculate angles between the (0001) and selected (hkil) planes. Metal HCP c/a Ratio ( 101 1) θ degrees ( 1121) θ degrees ( 101 2) θ degrees ( 1122) θ degrees ( 101 3) θ degrees ( 2023) θ degrees Be 1.56803 61.09 72.31 42.15 57.47 31.11 50.36 Mg 1.6235 61.92 72.88 43.15 58.37 32.00 51.34 Sc 1.59215 61.46 72.57 42.59 57.87 31.50 50.79 Ti 1.58734 61.38 72.52 42.50 57.79 31.42 50.70 Co 1.62283 61.91 72.88 43.14 58.36 31.99 51.32 Zn 1.85635 64.99 74.93 46.98 61.69 35.55 55.02 Y 1.56986 61.12 72.33 42.19 57.50 31.14 50.39 Zr 1.59312 61.47 72.58 42.61 57.88 31.52 50.81 Ru 1.5833 61.32 72.47 42.43 57.72 31.36 50.63 Cd 1.88572 65.33 75.15 47.43 62.06 35.97 55.44 Gd 1.58791 61.39 72.52 42.51 57.80 31.43 50.71 Tb 1.58056 61.28 72.45 42.38 57.68 31.31 50.58 Dy 1.57382 61.18 72.38 42.26 57.57 31.21 50.46 Ho 1.56983 61.12 72.33 42.19 57.50 31.14 50.39 Er 1.52877 60.47 71.89 41.43 56.81 30.47 49.64 Tm 1.57932 61.26 72.43 42.36 57.66 31.29 50.56 Lu 1.57143 61.14 72.35 42.22 57.53 31.17 50.42 Hf 1.58147 61.29 72.46 42.40 57.69 31.33 50.60 Re 1.61522 61.80 72.80 43.00 58.24 31.87 51.19 Os 1.57993 61.27 72.44 42.37 57.67 31.30 50.57 Tl 1.59821 61.55 72.63 42.70 57.97 31.60 50.90 Metal DHCP c/a Ratio ( 101 2) θ degrees ( 1122) θ degrees ( 101 4) θ degrees ( 1124) θ degrees ( 101 6) θ degrees ( 2026) θ degrees La 3.22546 61.76 72.77 42.96 58.20 31.83 51.15 Pr 3.22616 61.77 72.78 42.96 58.20 31.84 51.15 Nd 3.22404 61.75 72.77 42.94 58.19 31.82 51.14 Examination of Tables 4 and 5 indicates that in most cases the intermediate minimum values of E for HCP structures occur on planes inclined at angles of ~50 o to 60 o with respect to the (0001) plane, corresponding to a mix of planes of the type ( 2023) , ( 1122) and ( 101 1) . In contrast, intermediate maximum values of G occur at angles of approximately 45 o ±2 o corresponding to planes of the type 101 2) 472 ( . In the case of the three DCHP metals La, Pr and Nd, the minimum values of E are between ~48 o and ~56 o , near 50 o , corresponding to planes near 2026) maximum G values occur at angles near 46 o corresponding to planes near 101 4) ( . ( and the 3.6 Polar Diagrams of E and G. The θ versus E and θ versus G graphs are useful for comparing angular variations in moduli, but polar co-ordinate plots are more effective for assessing angular symmetry. Voigt [1928] and Wooster [1949] used compliance (S) polar plots for the mineral Beryl and Zinc. The present study uses polar plots of the modulus (i.e. reciprocal compliance S -1 ) where E is treated as a vector with co-ordinates E sin normal to the X3 axis and co-ordinates E cos parallel to the X3 axis. The shear modulus G is treated in the same manner. Importantly, each vector (modulus) is normalized with respect to the modulus value at θ = 90 o (i.e. sin 1 ) so that metals having very different moduli may be compared in the same figure. If all crystals were perfectly symmetrical in their modulus behaviour all polar plots would be circles with a radius of unity. Deviations from circular behaviour readily indicate the degree and direction of anisotropic behaviour. as shown in Fig, 9 for Mg, Cd, Zn and Be, and Ti, Y, Co and Zr. Metal groupings in this and subsequent figures were chosen to minimise overlap of individual data sets.
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