By Wolfgang Kunze
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The same year Nye's Physical Properties of Crystals (94) was published, which included a section containing expressions for the reciprocal of Young's modulus as a function of crystallographic orientation for the various crystal systems. For a thorough derivation and discussion of these formulas and similar ones for the reciprocal of the shear modulus the reader should consult Voigt's Lehrbuch der Kristallphysik (78). In his review article in 1958, Huntington (20) discussed briefly the relation between the elastic constants of single crystals and polycrystalline materials, commented on the difficulty associated with prediction of polycrystalline elastic constants from single crystal data, and summarized the work in this area.
S = c o s - 1 (la + mb + nc). (141) Substituting / = m = n = l/\/3 and a, 6, c from Eqs. (140) into Eq. -11-^-2^ . V 2 ( c 1 1 - c 1 2 + c 44 ) (143) Hence we see that the angle 5 by which the energy-flux vector deviates from the wave normal is dependent upon the particular values of the elastic constants for the material under investigation. In order to obtain more information about the energy-flux vector for the transverse waves with wave normal along [111] it is convenient to transform coordinates from *i = [100], x2 = [010], x3 = [001] (144) x2' = [T10], x3' = [111].
For the transverse waves v2 and i;3, a + /? , particle displacement along [IlO]. Thus, Eqs. (134) become £i= £ 2 3 co2A02[(cll-cl2)~] ^^;L—2—J 2>/3t = o2["(cii-c12)"1 i ^ | — --I 2 J 2^/3 »2L "2V3V44)' (138) 26 II. LINEAR ELASTIC WAVES and using Eq. (72) Using Eqs. (71), a = b = —- V2(c n -g 12 ) , 0 . ^ 0 ^ 1 / 2[(cn~c12)2 + 2 ^ > ' c = V2c44 [(Cll-cl2)2 + 2c2j* ' (140) The angle 5 by which the energy-flux deviates from the wave normal may be found from the relation. S = c o s - 1 (la + mb + nc).